Preliminary Question

#$&*

course Mth 173

1/11 10

-Answer (qa) Questions________________________________________

There are 2 questions in this document, accompanied by some instructions.

Copy this document into a word processor or text editor.

Answer the two questions posed in this document, inserting your answers, confidence assessments and self-critiques as explained.

• Solutions are given, but don't look at the solution to a question or problem until you have entered your answer. Preliminary Question

• You will probably find that you can answer many of these questions without writing anything down.

• It is often helpful to sketch, doodle, jot down ideas, do calculations, organize and test ideas on paper. On those problems where you cannot arrive at an answer 'in your head', is recommended that you work out your solutions on paper.

When appropriate, you will later be encouraged to use a calculator to do any arithmetic you cannot do mentally. However the calculator is not appropriate to the questions that appear on this document. Put the calculator aside and think through these questions.

When you are finished you will submit your work according to the instructions at the end of this document.

________________________________________

It is important that all the information in documents of this nature be submitted, so that all the information ends up posted at your page.

When submitting documents, do not delete anything from the original document. Insert your answers, questions, comments, etc. as indicated, but do not otherwise change the original document.

________________________________________

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Question: `q001. If you are earning money at the rate of 8 dollars / hour and work for 4 hours, how much money do you make during this time? Answer in such a way as to explain your reasoning as fully as possible. A solution to this problem appears several lines below, but enter your own solution before you look at the given solution.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

What I would do to get the solution is to multiply the rate of money made ( 8 dollars ) by how many hours worked ( 4 hours ). The answer would be 32 dollars an hour.

confidence rating #$&*: (Type in a number from 0 to 3, indicating your level of confidence in your solution.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

3 is my confidence.

.............................................

Given Solution: 8 dollars / hour means '8 dollars per hour', indicating that for every hour you work you earn 8 dollars. If you work for 4 hours, then if you earn 8 dollars for every one of those hours you earn 4 * 8 dollars = 32 dollars.

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Self-critique (if necessary): (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

ok

------------------------------------------------

Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

*********************************************

Question: `q002. If you work 12 hours and earn $168, then at what rate, in dollars / hour, were you making money?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

To find the solution I would divide the money earned ( 168 dollars ) by the hours worked. That would make the answer 14 dollars an hour.

confidence rating #$&*: (Type in a number from 0 to 3, indicating your level of confidence in your solution.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

3

.............................................

Given Solution: $168 earned in 12 hours implies that $168 / 12 = $14 were made per hour, so the rate is $14 / hour.

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Self-critique (if necessary): (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

ok

------------------------------------------------

Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique, using a number between 0 and 3.

3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

________________________________________

There are 10 more questions, but before you work them you should see how to submit your work. So take a couple of minutes and submit a copy of the everything above this point in your document, using the Submit Work Form at http://vhcc2.vhcc.edu/dsmith/submit_work.htm. The form has instructions but read the following:

• You will be asked to give your work a title. You may use any title you wish; if you aren't sure what you want to call it, just call it 'First Two Questions' or something of that nature. The title you choose is the title under which your work will be posted after the instructor has reviewed it.

• You will simply copy and paste everything that precedes this paragraph, including your answers, confidence assessments, self-critiques, etc., into a box in the form, and click Submit.

• Your work will then be posted by the end of the following day, and often by the end of the day on which you submit it, at your personal access site. You received instructions for accessing this site with your access code.

It is suggested that you bookmark the Submit Work Form now, but if you don't you will be reminded later.

"

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

course Mth 173

1/11 10

Introductory Question-Answer (qa) Sequence________________________________________

There are 12 questions in this document, along with some instructions.

Copy this document into a word processor or text editor.

Answer the first two questions below, inserting your answers, Confidence Ratings and self-critiques as explained. You will then read the instructions that precede the remaining ten questions and answer those questions in a similar manner. Finally you will submit your work using a web form, according to instructions at the end of the document.

You will probably find that you can answer many of these questions without writing anything down. On those problems where you cannot arrive at an answer 'in your head', is recommended that you work out your solutions on paper. It is often helpful to sketch, doodle, jot down ideas, do calculations, organize and test ideas on paper.

When appropriate, you will later be encouraged to use a calculator to do any arithmetic you cannot do mentally. However the calculator is not appropriate to the questions that appear on this document. Put the calculator aside and think through these questions.

________________________________________

Here is some additional information on the process and how it will fit into your course:

One of the predominant features of your course is the question-answer format for submitting work.

• In most courses you will encounter sequenced questions of this nature, designed to build your understanding by engaging you in the process of answering and self-critiquing your answers on a number of questions.

• In all courses you will submit assigned problems using this format.

As with the first couple of questions, the questions in this document can be answered with just a knowledge of basic high-school mathematics.

Sometimes the given solutions are more subtle than you might expect, and you will probably find that many of your answers, while good and correct, do not completely match the given solution. This is intentional, the goal being to get you used to the idea and the benefits of the self-critique process.

• Don't worry if you have trouble with a few of the questions, or if your explanations don't quite match those in the given solutions--most students begin their course a little rusty.

• Be sure to do your best to understand all the questions and the given solutions--it's this effort that makes the process beneficial to you.

The process is fairly simple, and you'll be using it again and again.

• The process will soon be very familiar to you, if it isn't already.

• Work through the instructions given here and within the questions, and do your best.

• If you miss something in the process (as most students are bound to do the first time through), your instructor will point it out to you, and there will be ample opportunity to get everything straight.

________________________________________

Your basic instructions follow. Rather than giving you the instructions at the beginning of the document, you were given a couple of questions to serve as a point of reference, and should now be better prepared to understand the instructions:

1. Answer each question, then look at the given solution:

It is expected that you will answer each question before looking at the given solution. There is no grade penalty for looking ahead, but if you do you:

• may be bypassing an opportunity to engage yourself in the solution process

• run the risk of deluding yourself about what you understand

• are likely to learn much less and

• are not as likely to do well in the course.

However your instructor understands the tradeoffs involved in being a student, and makes no judgement about how you should use this material. As long as you use it to your best advantage and succeed on tests and other course activities, you will get a good result from this course.

2. If you can't readily work it out in your head, use pencil and paper, and keep a record of your work.

You aren't expected to work out your solutions by staring at a computer screen, though you will likely find many questions and problems easy enough to do 'in your head'. However on more challenging problems, it's easier to work things out using a handwritten document than a computer-created or word-processed document.

• You should in general work out your answers to non-obvious problems on paper, jotting down sketches, diagrams and notes as you go, in such a way that you can make sense of it later. This will help you focus your work and maintain your train of thought, and will be quite valuable for periodic review. It is recommended (and may in some courses be required) that you dedicate a notebook to this course, and at least sketch out your work in the notebook.

3. There's no need for special formatting or graphs:

• Don't use special characters in your responses (e.g., characters like  that don't appear on your keyboard). The characters on your keyboard suffice to answer all these questions. If you use special characters they won't come through the form you use to submit your work, and if you use too many such characters your instructor might not be able to tell what you are saying.

• Don't try to make graphs in your document. Sketch your graphs by hand, then if necessary describe them in words (that probably won't be necessary in the present exercise; more about that later). Graphs won't come through when you submit your work. You can make a graph without understanding it, but you can't give a good description of your graph without understanding it. Your instructor doesn't need to see your graphs; he needs to see your descriptions of your graphs. The present exercise doesn't require extensive descriptions of graphs; it they apply to your course, you will see more about describing your graphs soon.

• When you submit this document (per instructions at the end), it will come to the instructor in pure text format. Any formatting you have done will not be seen by the instructor, special characters will not appear in what the instructor sees, and graphs won't come through. So don't do any fancy formatting for the instructor. You will of course want to save your original copy, and you are welcome to add formatting for your own purposes.

4. Expect to see your work posted by the instructor in a timely manner:

After submitting your work, per instructions at the end of this document, you should expect to see your work posted, along with instructor commentary, at your access page. It should be posted by the evening of the day after you submit it, and may well be posted the evening of the same day.

You have probably submitted your answers the first two questions in a preceding task. You are welcome to answer them again, but if you have already submitted them you may go ahead and skip to the third question.

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Question: `q001. If you are earning money at the rate of 8 dollars / hour and work for 4 hours, how much money do you make during this time? Answer in such a way as to explain your reasoning as fully as possible. A solution to this problem appears several lines below, but enter your own solution before you look at the given solution.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

What I would do to get the solution is to multiply the rate of money made ( 8 dollars ) by how many hours worked ( 4 hours ). The answer would be 32 dollars an hour.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: 8 dollars / hour means '8 dollars per hour', indicating that for every hour you work you earn 8 dollars. If you work for 4 hours, then if you earn 8 dollars for every one of those hours you earn 4 * 8 dollars = 32 dollars.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

ok

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

*********************************************

Question: `q002. If you work 12 hours and earn $168, then at what rate, in dollars / hour, were you making money?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

To find the solution I would divide the money earned ( 168 dollars ) by the hours worked. That would make the answer 14 dollars an hour.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: $168 earned in 12 hours implies that $168 / 12 = $14 were made per hour, so the rate is $14 / hour.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

ok

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

________________________________________

________________________________________

Here are the remaining ten questions:

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Question: `q003. If you are earning 8 dollars / hour, how long will it take you to earn $72? The answer may well be obvious, but explain as best you can how you reasoned out your result.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

Divide $72 by 8 dollars so the hours worked will show. The answer is 9 hours worked.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

.............................................

Given Solution: Many students simply know, at the level of common sense, that if we divide $72 by $8 / hour we get 9 hours, so 9 hours are required.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

ok

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

*********************************************

Question: `q004. Calculate (8 + 3) * 5 and 8 + 3 * 5, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

Order of operations.

(8 + 3) * 5

First parenthesis (8 + 3) = 11

Second times 11 * 5 = 55

Solution = 55

8 + 3 * 5

First times 3 * 5 = 15

Second add 8 + 15 = 23

The reason the answers are different are because the way they are wrote. One has parenthesis the other does not.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: (8 + 3) * 5 and 8 + 3 * 5

To evaluate (8 + 3) * 5, you will first do the calculation in parentheses. 8 + 3 = 11, so

(8 + 3) * 5 = 11 * 5 = 55.

To evaluate 8 + 3 * 5 you have to decide which operation to do first, 8 + 3 or 3 * 5. You should be familiar with the order of operations, which tells you that multiplication precedes addition. The first calculation to do is therefore 3 * 5, which is equal to 15. Thus

8 + 3 * 5 = 8 + 15 = 23

The results are different because the grouping in the first expression dictates that the addition be done first.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

ok

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

In subsequent problems the detailed instructions that accompanied the first four problems are missing. We assume you will know to follow the same instructions in answering the remaining questions.

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Question: `q005. Calculate (2^4) * 3 and 2^(4 * 3), indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. Note that the symbol '^' indicates raising to a power. For example, 4^3 means 4 raised to the third power, which is the same as 4 * 4 * 4 = 64.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Order of operations

(2^4) * 3

First parenthesis 2 to the fourth power = 16

Second times 16 * 3 = 48

2^(4 * 3)

First parenthesis (4*3) = 12

Second powers 2^12 = 4096

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: 3

To evaluate (2^4) * 3 we first evaluate the grouped expression 2^4, which is the fourth power of 2, equal to 2 * 2 * 2 * 2 = 16. So we have

(2^4) * 3 = 16 * 3 = 48.

To evaluate 2^(4 * 3) we first do the operation inside the parentheses, obtaining 4 * 3 = 12. We therefore get

2^(4 * 3) = 2^12 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 4096.

It is easy to multiply by 2, and the powers of 2 are important, so it's appropriate to have asked you to do this problem without using a calculator. Had the exponent been much higher, or had the calculation been, say, 3^12, the calculation would have become tedious and error-prone, and the calculator would have been recommended.

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Self-critique (if necessary):

ok

*********************************************

Question: `q006. Calculate 3 * 5 - 4 * 3 ^ 2 and 3 * 5 - (4 * 3)^2 according to the standard order of operations, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

3 * 5 – 4 * 3 ^ 2

First powers 3 ^ 2 = 9

Second multiply 3 * 5 = 15 and 4 * 9 = 36

Third subtract 15 – 36 = -21

3 * 5 – (4 * 3) ^ 2

First parenthesis (4 * 3) = 12

Second powers 12 ^ 2 = 144

Third multiply 3 * 5 = 15

Fourth subtract 15 – 144 = 129

confidence rating #$&*:

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Given Solution:

To calculate 3 * 5 - 4 * 3 ^ 2, the first operation is the exponentiation operation ^.

• The two numbers involved in the exponentiation are 3 and 2; the 4 is 'attached' to the 3 by multiplication, and this multiplication can't be done until the exponentiation has been performed.

• The exponentiation operation is therefore 3^2 = 9, and the expression becomes 3 * 5 - 4 * 9.

Evaluating this expression, the multiplications 3 * 5 and 4 * 9 must be performed before the subtraction. 3 * 5 = 15 and 4 * 9 = 36 so we now have

3 * 5 - 4 * 3 ^ 2 = 3 * 5 - 4 * 9 = 15 - 36 = -21.

To calculate 3 * 5 - (4 * 3)^2 we first do the operation in parentheses, obtaining 4 * 3 = 12. Then we apply the exponentiation to get 12 ^2 = 144. Finally we multiply 3 * 5 to get 15. Putting this all together we get

3 * 5 - (4 * 3)^2 =

3 * 5 - 12^2 =

3 * 5 - 144 =

15 - 144 =

-129.

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Self-critique (if necessary):

ok

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Self-critique Rating: 3

In the next three problems, the graphs will be of one of the basic shapes listed below. You will be asked to construct graphs for three simple functions, and determine which of the depicted graphs each of your graphs most closely resembles. At this point you won't be expected to know these terms or these graph shapes; if at some point in your course you are expected to know these things, they will be presented at that point.

Linear:

Quadratic or parabolic:

Exponential:

Odd power:

Fractional positive power:

Even negative power:

partial graph of polynomial of degree 3

more extensive graph of polynomial of degree 3

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Question: `q007. Let y = 2 x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

-2

-1

0

1

2

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

y = 2 x + 3.

For -2 = 2(-2) +3

-4 + 3 = -1

For -1 = 2 (-1) + 3

-2 + 3 = 1

For 0 = 2(0) + 3

0 + 3 = 3

For 1 = 2(1) + 3

2 + 3 = 5

For 2 = 2(2) + 3

4 + 3 = 7

The graph is a linear graph that goes through the points of (-2,-1), (-1,1), (0, 3), (1, 5), (2,7). It crosses the y axis at 3.

confidence rating #$&*:

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Given Solution:

Two slightly different explanations are give below, one by a student and one by the instructor. Neither format is inherently better than the other.

GOOD SOLUTION BY STUDENT:

First we need to complete the table. I have added a column to the right of the table to show the calculation of “y” when we us the “x” values as given.

x y Calculation: If y = 2x + 3

-2 -1 If x = -2, then y = 2(-2)+3 = -4+3 = -1

-1 1 If x= -1, then y = 2(-1)+3 = -2+3 = 1

0 3 If x= 0, then y = 2(0)+3 = 0+3 = 3

1 5 If x= 1, then y = 2(1)+3 = 2+3 = 5

2 7 If x= 2, then y = 2(2)+3 = 4+3 = 7

Once an answer has been determined, the “y” value can be filled in. Now we have both the “x” and “y” values and we can begin our graph. The charted values continue on a straight line representing a linear function as shown above.

INSTRUCTOR'S SOLUTION:

We easily evaluate the expression:

• When x = -2, we get y = 2 x + 3 = 2 * (-2) + 3 = -4 + 3 = -1.

• When x = -1, we get y = 2 x + 3 = 2 * (-1) + 3 = -2 + 3 = 1.

• When x = 0, we get y = 2 x + 3 = 2 * (0) + 3 = 0 + 3 = 3.

• When x = 1, we get y = 2 x + 3 = 2 * (1) + 3 = 2 + 3 = 5.

• When x = 2, we get y = 2 x + 3 = 2 * (2) + 3 = 4 + 3 = 7.

Filling in the table we have

x y

-2 -1

-1 1

0 3

1 5

2 7

When we graph these points we find that they lie along a straight line.

Only one of the depicted graphs consists of a straight line, and we conclude that the appropriate graph is the one labeled 'linear'.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `q008. Let y = x^2 + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

-2

-1

0

1

2

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

y = x^2 + 3

x y Calculation: If y = x^2 + 3

-2 7 If x = -2, then y = -2^2 + 3 = 4 + 3 = 7

-1 4 If x= -1, then y = -1^2 + 3 = 1 + 3 = 4

0 3 If x= 0, then y = 0^2 + 3 = 0+3 = 3

1 4 If x= 1, then y = 1^2 + 3 = 4

2 7 If x= 2, then y = 2^2 + 3 = 7

The graph would be a parabola with the vertex of (0, 3) and facing up.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Evaluating y = x^2 + 3 at the five points:

• If x = -2 then we obtain y = x^2 + 3 = (-2)^2 + 3 = 4 + 3 = 7.

• If x = -1 then we obtain y = x^2 + 3 = (-1)^2 + 3 = ` + 3 = 4.

• If x = 0 then we obtain y = x^2 + 3 = (0)^2 + 3 = 0 + 3 = 3.

• If x = 1 then we obtain y = x^2 + 3 = (1)^2 + 3 = 1 + 3 = 4.

• If x = 2 then we obtain y = x^2 + 3 = (2)^2 + 3 = 4 + 3 = 7.

The table becomes

x y

-2 7

-1 4

0 3

1 4

2 7

We note that there is a symmetry to the y values. The lowest y value is 3, and whether we move up or down the y column from the value 3, we find the same numbers (i.e., if we move 1 space up from the value 3 the y value is 4, and if we move one space down we again encounter 4; if we move two spaces in either direction from the value 3, we find the value 7).

A graph of y vs. x has its lowest point at (0, 3).

If we move from this point, 1 unit to the right our graph rises 1 unit, to (1, 4), and if we move 1 unit to the left of our 'low point' the graph rises 1 unit, to (-1, 4).

If we move 2 units to the right or the left from our 'low point', the graph rises 4 units, to (2, 7) on the right, and to (-2, 7) on the left.

Thus as we move from our 'low point' the graph rises up, becoming increasingly steep, and the behavior is the same whether we move to the left or right of our 'low point'. This reflects the symmetry we observed in the table. So our graph will have a right-left symmetry.

Two of the depicted graphs curve upward away from the 'low point'. One is the graph labeled 'quadratic or parabolic'. The other is the graph labeled 'partial graph of degree 3 polynomial'.

If we look closely at these graphs, we find that only the first has the right-left symmetry, so the appropriate graph is the 'quadratic or parabolic' graph.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `q009. Let y = 2 ^ x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = 1. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values 2, 3 and 4. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

1

2

3

4

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

x y Calculation: If y = 2 ^ x + 3

1 5 If x = 1, then y = 2 ^ 1 + 3 = 2 + 3 = 5

2 7 If x= 2, then y = 2 ^ 2 + 3 = 4 + 3 =7

3 11 If x= 3, then y = 2 ^ 3 + 3 = 11

4 19 If x= 4, then y = 2 ^ 4 + 3 = 16 + 3 =19

The graph would be exponential going upwards with the initial point of (1, 5).

confidence rating #$&*:

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.............................................

Given Solution:

Recall that the exponentiation in the expression 2^x + 1 must be done before, not after the addition.

When x = 1 we obtain y = 2^1 + 3 = 2 + 3 = 5.

When x = 2 we obtain y = 2^2 + 3 = 4 + 3 = 7.

When x = 3 we obtain y = 2^3 + 3 = 8 + 3 = 11.

When x = 4 we obtain y = 2^4 + 3 = 16 + 3 = 19.

x y

1 5

2 7

3 11

4 19

Looking at the numbers in the y column we see that they increase as we go down the column, and that the increases get progressively larger. In fact if we look carefully we see that each increase is double the one before it, with increases of 2, then 4, then 8.

When we graph these points we find that the graph rises as we go from left to right, and that it rises faster and faster. From our observations on the table we know that the graph in fact that the rise of the graph doubles with each step we take to the right.

The only graph that increases from left to right, getting steeper and steeper with each step, is the graph labeled 'exponential'.

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Self-critique (if necessary):

In this graph there is work that I had to plug in to get the graph. I do not look at this type of math everyday. I could not tell what kind of graph it was so I had to do the work and graph it out on a separate piece of paper to reveal what it was.

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Self-critique Rating:

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Question: `q010. If you divide a certain positive number by 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

It would be equal to the original number because any number divided by one is always going to be equal to the original number.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: If you divide any number by 1, the result is the same as the original number. Doesn't matter what the original number is, if you divide it by 1, you don't change it.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `q011. If you divide a certain positive number by a number greater than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The answer would be less than the original number.

Example:

3 / 2 = 1.5

3 / 6 = .5

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by another number is similar. The bigger the number you divide by, the less you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a number greater than 1, what you get has to be smaller than the original number. Again it doesn't matter what the original number is, as long as it's positive.

Students will often reason from examples. For instance, the following reasoning might be offered:

OK, let's say the original number is 36. Let's divide 36 be a few numbers and see what happens:

36/2 = 18. Now 3 is bigger than 2, and

36 / 3 = 12. The quotient got smaller. Now 4 is bigger than 3, and

36 / 4 = 9. The quotient got smaller again. Let's skip 5 because it doesn't divide evenly into 36.

36 / 6 = 4. Again we divided by a larger number and the quotient was smaller.

I'm convinced.

That is a pretty convincing argument, mainly because it is so consistent with our previous experience. In that sense it's a good argument. It's also useful, giving us a concrete example of how dividing by bigger and bigger numbers gives us smaller and smaller results.

However specific examples, however convincing and however useful, don't actually prove anything. The argument given at the beginning of this solution is general, and applies to all positive numbers, not just the specific positive number chosen here.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `q012. If you divide a certain positive number by a positive number less than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The answer is the result would be greater than the original number.

Example:

12 / .5 = 24

12/.3 = 40

confidence rating #$&*:

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Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by some other number is similar. The bigger the number you divide by, the less you get. The smaller the number you divide by, the more you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a positive number less than 1, what you get has to be larger than the original number. Again it doesn't matter what the original number is, as long as it's positive.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `q013. Students often get the basic answers to nearly all, or even all these questions, correct. Your instructor has however never seen anyone who addressed all the subtleties in the given solutions in their self-critiques, and it is very common for a student to have given no self-critiques. It is very likely that there is something in the given solutions that is not expressed in your solution.

This doesn't mean that you did a bad job. If you got most of the 'answers' right, you did fine.

However, in order to better understand the process, you are asked here to go back and find something in one of the given solutions that you did not address in your solution, and insert a self-critique. You should choose something that isn't trivial to you--something you're not 100% sure you understand.

If you can't find anything, you can indicate this below, and the instructor will point out something and request a response (the instructor will select something reasonable, but will then expect a very good and complete response). However it will probably be less work for you if you find something yourself.

Your response should be inserted at the appropriate place in this document, and should be indicated by preceding it with ####.

As an answer to this question, include a copy of whatever you inserted above, or an indication that you can't find anything.

your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

________________________________________

Submit a copy of this document using the Submit Work Form at http://vhcc2.vhcc.edu/dsmith/submit_work.htm. The form has instructions but read the following:

• You will be asked to give your work a title. You may use any title you wish; if you aren't sure what you want to call it, just call it 'First Two Questions' or something of that nature. The title you choose is the title under which your work will be posted after the instructor has reviewed it.

• You will simply copy and paste everything that precedes this paragraph, including your answers, Confidence Ratings, self-critiques, etc., into a box in the form, and click Submit.

• Your work will then be posted by the end of the following day, and often by the end of the day on which you submit it, at your personal access site. You received instructions for accessing this site with your access code.

It is suggested that you bookmark the Submit Work Form now, but if you don't you will be reminded later.

________________________________________

When you have submitted this document, you will have complete Step 3 of the 8-step Orientation. Your next step will be to return tohttp://vhcc2.vhcc.edu/dsmith/geninfo/startup_and_orientation.htm and continue on to Step 4.

"

Self-critique (if necessary):

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Self-critique rating:

________________________________________

&#Good responses. Let me know if you have questions. &#

#$&*

course Mth 173

1/11 10

Introductory Question-Answer (qa) Sequence________________________________________

There are 12 questions in this document, along with some instructions.

Copy this document into a word processor or text editor.

Answer the first two questions below, inserting your answers, Confidence Ratings and self-critiques as explained. You will then read the instructions that precede the remaining ten questions and answer those questions in a similar manner. Finally you will submit your work using a web form, according to instructions at the end of the document.

You will probably find that you can answer many of these questions without writing anything down. On those problems where you cannot arrive at an answer 'in your head', is recommended that you work out your solutions on paper. It is often helpful to sketch, doodle, jot down ideas, do calculations, organize and test ideas on paper.

When appropriate, you will later be encouraged to use a calculator to do any arithmetic you cannot do mentally. However the calculator is not appropriate to the questions that appear on this document. Put the calculator aside and think through these questions.

________________________________________

Here is some additional information on the process and how it will fit into your course:

One of the predominant features of your course is the question-answer format for submitting work.

• In most courses you will encounter sequenced questions of this nature, designed to build your understanding by engaging you in the process of answering and self-critiquing your answers on a number of questions.

• In all courses you will submit assigned problems using this format.

As with the first couple of questions, the questions in this document can be answered with just a knowledge of basic high-school mathematics.

Sometimes the given solutions are more subtle than you might expect, and you will probably find that many of your answers, while good and correct, do not completely match the given solution. This is intentional, the goal being to get you used to the idea and the benefits of the self-critique process.

• Don't worry if you have trouble with a few of the questions, or if your explanations don't quite match those in the given solutions--most students begin their course a little rusty.

• Be sure to do your best to understand all the questions and the given solutions--it's this effort that makes the process beneficial to you.

The process is fairly simple, and you'll be using it again and again.

• The process will soon be very familiar to you, if it isn't already.

• Work through the instructions given here and within the questions, and do your best.

• If you miss something in the process (as most students are bound to do the first time through), your instructor will point it out to you, and there will be ample opportunity to get everything straight.

________________________________________

Your basic instructions follow. Rather than giving you the instructions at the beginning of the document, you were given a couple of questions to serve as a point of reference, and should now be better prepared to understand the instructions:

1. Answer each question, then look at the given solution:

It is expected that you will answer each question before looking at the given solution. There is no grade penalty for looking ahead, but if you do you:

• may be bypassing an opportunity to engage yourself in the solution process

• run the risk of deluding yourself about what you understand

• are likely to learn much less and

• are not as likely to do well in the course.

However your instructor understands the tradeoffs involved in being a student, and makes no judgement about how you should use this material. As long as you use it to your best advantage and succeed on tests and other course activities, you will get a good result from this course.

2. If you can't readily work it out in your head, use pencil and paper, and keep a record of your work.

You aren't expected to work out your solutions by staring at a computer screen, though you will likely find many questions and problems easy enough to do 'in your head'. However on more challenging problems, it's easier to work things out using a handwritten document than a computer-created or word-processed document.

• You should in general work out your answers to non-obvious problems on paper, jotting down sketches, diagrams and notes as you go, in such a way that you can make sense of it later. This will help you focus your work and maintain your train of thought, and will be quite valuable for periodic review. It is recommended (and may in some courses be required) that you dedicate a notebook to this course, and at least sketch out your work in the notebook.

3. There's no need for special formatting or graphs:

• Don't use special characters in your responses (e.g., characters like  that don't appear on your keyboard). The characters on your keyboard suffice to answer all these questions. If you use special characters they won't come through the form you use to submit your work, and if you use too many such characters your instructor might not be able to tell what you are saying.

• Don't try to make graphs in your document. Sketch your graphs by hand, then if necessary describe them in words (that probably won't be necessary in the present exercise; more about that later). Graphs won't come through when you submit your work. You can make a graph without understanding it, but you can't give a good description of your graph without understanding it. Your instructor doesn't need to see your graphs; he needs to see your descriptions of your graphs. The present exercise doesn't require extensive descriptions of graphs; it they apply to your course, you will see more about describing your graphs soon.

• When you submit this document (per instructions at the end), it will come to the instructor in pure text format. Any formatting you have done will not be seen by the instructor, special characters will not appear in what the instructor sees, and graphs won't come through. So don't do any fancy formatting for the instructor. You will of course want to save your original copy, and you are welcome to add formatting for your own purposes.

4. Expect to see your work posted by the instructor in a timely manner:

After submitting your work, per instructions at the end of this document, you should expect to see your work posted, along with instructor commentary, at your access page. It should be posted by the evening of the day after you submit it, and may well be posted the evening of the same day.

You have probably submitted your answers the first two questions in a preceding task. You are welcome to answer them again, but if you have already submitted them you may go ahead and skip to the third question.

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Question: `q001. If you are earning money at the rate of 8 dollars / hour and work for 4 hours, how much money do you make during this time? Answer in such a way as to explain your reasoning as fully as possible. A solution to this problem appears several lines below, but enter your own solution before you look at the given solution.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

What I would do to get the solution is to multiply the rate of money made ( 8 dollars ) by how many hours worked ( 4 hours ). The answer would be 32 dollars an hour.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: 8 dollars / hour means '8 dollars per hour', indicating that for every hour you work you earn 8 dollars. If you work for 4 hours, then if you earn 8 dollars for every one of those hours you earn 4 * 8 dollars = 32 dollars.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

ok

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Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

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Question: `q002. If you work 12 hours and earn $168, then at what rate, in dollars / hour, were you making money?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

To find the solution I would divide the money earned ( 168 dollars ) by the hours worked. That would make the answer 14 dollars an hour.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

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Given Solution: $168 earned in 12 hours implies that $168 / 12 = $14 were made per hour, so the rate is $14 / hour.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

ok

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

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Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

________________________________________

________________________________________

Here are the remaining ten questions:

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Question: `q003. If you are earning 8 dollars / hour, how long will it take you to earn $72? The answer may well be obvious, but explain as best you can how you reasoned out your result.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

Divide $72 by 8 dollars so the hours worked will show. The answer is 9 hours worked.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

.............................................

Given Solution: Many students simply know, at the level of common sense, that if we divide $72 by $8 / hour we get 9 hours, so 9 hours are required.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

ok

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

*********************************************

Question: `q004. Calculate (8 + 3) * 5 and 8 + 3 * 5, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

Order of operations.

(8 + 3) * 5

First parenthesis (8 + 3) = 11

Second times 11 * 5 = 55

Solution = 55

8 + 3 * 5

First times 3 * 5 = 15

Second add 8 + 15 = 23

The reason the answers are different are because the way they are wrote. One has parenthesis the other does not.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: (8 + 3) * 5 and 8 + 3 * 5

To evaluate (8 + 3) * 5, you will first do the calculation in parentheses. 8 + 3 = 11, so

(8 + 3) * 5 = 11 * 5 = 55.

To evaluate 8 + 3 * 5 you have to decide which operation to do first, 8 + 3 or 3 * 5. You should be familiar with the order of operations, which tells you that multiplication precedes addition. The first calculation to do is therefore 3 * 5, which is equal to 15. Thus

8 + 3 * 5 = 8 + 15 = 23

The results are different because the grouping in the first expression dictates that the addition be done first.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

ok

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

In subsequent problems the detailed instructions that accompanied the first four problems are missing. We assume you will know to follow the same instructions in answering the remaining questions.

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Question: `q005. Calculate (2^4) * 3 and 2^(4 * 3), indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. Note that the symbol '^' indicates raising to a power. For example, 4^3 means 4 raised to the third power, which is the same as 4 * 4 * 4 = 64.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Order of operations

(2^4) * 3

First parenthesis 2 to the fourth power = 16

Second times 16 * 3 = 48

2^(4 * 3)

First parenthesis (4*3) = 12

Second powers 2^12 = 4096

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: 3

To evaluate (2^4) * 3 we first evaluate the grouped expression 2^4, which is the fourth power of 2, equal to 2 * 2 * 2 * 2 = 16. So we have

(2^4) * 3 = 16 * 3 = 48.

To evaluate 2^(4 * 3) we first do the operation inside the parentheses, obtaining 4 * 3 = 12. We therefore get

2^(4 * 3) = 2^12 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 4096.

It is easy to multiply by 2, and the powers of 2 are important, so it's appropriate to have asked you to do this problem without using a calculator. Had the exponent been much higher, or had the calculation been, say, 3^12, the calculation would have become tedious and error-prone, and the calculator would have been recommended.

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Self-critique (if necessary):

ok

*********************************************

Question: `q006. Calculate 3 * 5 - 4 * 3 ^ 2 and 3 * 5 - (4 * 3)^2 according to the standard order of operations, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

3 * 5 – 4 * 3 ^ 2

First powers 3 ^ 2 = 9

Second multiply 3 * 5 = 15 and 4 * 9 = 36

Third subtract 15 – 36 = -21

3 * 5 – (4 * 3) ^ 2

First parenthesis (4 * 3) = 12

Second powers 12 ^ 2 = 144

Third multiply 3 * 5 = 15

Fourth subtract 15 – 144 = 129

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

To calculate 3 * 5 - 4 * 3 ^ 2, the first operation is the exponentiation operation ^.

• The two numbers involved in the exponentiation are 3 and 2; the 4 is 'attached' to the 3 by multiplication, and this multiplication can't be done until the exponentiation has been performed.

• The exponentiation operation is therefore 3^2 = 9, and the expression becomes 3 * 5 - 4 * 9.

Evaluating this expression, the multiplications 3 * 5 and 4 * 9 must be performed before the subtraction. 3 * 5 = 15 and 4 * 9 = 36 so we now have

3 * 5 - 4 * 3 ^ 2 = 3 * 5 - 4 * 9 = 15 - 36 = -21.

To calculate 3 * 5 - (4 * 3)^2 we first do the operation in parentheses, obtaining 4 * 3 = 12. Then we apply the exponentiation to get 12 ^2 = 144. Finally we multiply 3 * 5 to get 15. Putting this all together we get

3 * 5 - (4 * 3)^2 =

3 * 5 - 12^2 =

3 * 5 - 144 =

15 - 144 =

-129.

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Self-critique (if necessary):

ok

------------------------------------------------

Self-critique Rating: 3

In the next three problems, the graphs will be of one of the basic shapes listed below. You will be asked to construct graphs for three simple functions, and determine which of the depicted graphs each of your graphs most closely resembles. At this point you won't be expected to know these terms or these graph shapes; if at some point in your course you are expected to know these things, they will be presented at that point.

Linear:

Quadratic or parabolic:

Exponential:

Odd power:

Fractional positive power:

Even negative power:

partial graph of polynomial of degree 3

more extensive graph of polynomial of degree 3

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Question: `q007. Let y = 2 x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

-2

-1

0

1

2

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

y = 2 x + 3.

For -2 = 2(-2) +3

-4 + 3 = -1

For -1 = 2 (-1) + 3

-2 + 3 = 1

For 0 = 2(0) + 3

0 + 3 = 3

For 1 = 2(1) + 3

2 + 3 = 5

For 2 = 2(2) + 3

4 + 3 = 7

The graph is a linear graph that goes through the points of (-2,-1), (-1,1), (0, 3), (1, 5), (2,7). It crosses the y axis at 3.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Two slightly different explanations are give below, one by a student and one by the instructor. Neither format is inherently better than the other.

GOOD SOLUTION BY STUDENT:

First we need to complete the table. I have added a column to the right of the table to show the calculation of “y” when we us the “x” values as given.

x y Calculation: If y = 2x + 3

-2 -1 If x = -2, then y = 2(-2)+3 = -4+3 = -1

-1 1 If x= -1, then y = 2(-1)+3 = -2+3 = 1

0 3 If x= 0, then y = 2(0)+3 = 0+3 = 3

1 5 If x= 1, then y = 2(1)+3 = 2+3 = 5

2 7 If x= 2, then y = 2(2)+3 = 4+3 = 7

Once an answer has been determined, the “y” value can be filled in. Now we have both the “x” and “y” values and we can begin our graph. The charted values continue on a straight line representing a linear function as shown above.

INSTRUCTOR'S SOLUTION:

We easily evaluate the expression:

• When x = -2, we get y = 2 x + 3 = 2 * (-2) + 3 = -4 + 3 = -1.

• When x = -1, we get y = 2 x + 3 = 2 * (-1) + 3 = -2 + 3 = 1.

• When x = 0, we get y = 2 x + 3 = 2 * (0) + 3 = 0 + 3 = 3.

• When x = 1, we get y = 2 x + 3 = 2 * (1) + 3 = 2 + 3 = 5.

• When x = 2, we get y = 2 x + 3 = 2 * (2) + 3 = 4 + 3 = 7.

Filling in the table we have

x y

-2 -1

-1 1

0 3

1 5

2 7

When we graph these points we find that they lie along a straight line.

Only one of the depicted graphs consists of a straight line, and we conclude that the appropriate graph is the one labeled 'linear'.

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Self-critique (if necessary):

------------------------------------------------

Self-critique Rating:

*********************************************

Question: `q008. Let y = x^2 + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

-2

-1

0

1

2

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

y = x^2 + 3

x y Calculation: If y = x^2 + 3

-2 7 If x = -2, then y = -2^2 + 3 = 4 + 3 = 7

-1 4 If x= -1, then y = -1^2 + 3 = 1 + 3 = 4

0 3 If x= 0, then y = 0^2 + 3 = 0+3 = 3

1 4 If x= 1, then y = 1^2 + 3 = 4

2 7 If x= 2, then y = 2^2 + 3 = 7

The graph would be a parabola with the vertex of (0, 3) and facing up.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Evaluating y = x^2 + 3 at the five points:

• If x = -2 then we obtain y = x^2 + 3 = (-2)^2 + 3 = 4 + 3 = 7.

• If x = -1 then we obtain y = x^2 + 3 = (-1)^2 + 3 = ` + 3 = 4.

• If x = 0 then we obtain y = x^2 + 3 = (0)^2 + 3 = 0 + 3 = 3.

• If x = 1 then we obtain y = x^2 + 3 = (1)^2 + 3 = 1 + 3 = 4.

• If x = 2 then we obtain y = x^2 + 3 = (2)^2 + 3 = 4 + 3 = 7.

The table becomes

x y

-2 7

-1 4

0 3

1 4

2 7

We note that there is a symmetry to the y values. The lowest y value is 3, and whether we move up or down the y column from the value 3, we find the same numbers (i.e., if we move 1 space up from the value 3 the y value is 4, and if we move one space down we again encounter 4; if we move two spaces in either direction from the value 3, we find the value 7).

A graph of y vs. x has its lowest point at (0, 3).

If we move from this point, 1 unit to the right our graph rises 1 unit, to (1, 4), and if we move 1 unit to the left of our 'low point' the graph rises 1 unit, to (-1, 4).

If we move 2 units to the right or the left from our 'low point', the graph rises 4 units, to (2, 7) on the right, and to (-2, 7) on the left.

Thus as we move from our 'low point' the graph rises up, becoming increasingly steep, and the behavior is the same whether we move to the left or right of our 'low point'. This reflects the symmetry we observed in the table. So our graph will have a right-left symmetry.

Two of the depicted graphs curve upward away from the 'low point'. One is the graph labeled 'quadratic or parabolic'. The other is the graph labeled 'partial graph of degree 3 polynomial'.

If we look closely at these graphs, we find that only the first has the right-left symmetry, so the appropriate graph is the 'quadratic or parabolic' graph.

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Self-critique (if necessary):

------------------------------------------------

Self-critique Rating:

*********************************************

Question: `q009. Let y = 2 ^ x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = 1. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values 2, 3 and 4. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

1

2

3

4

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

x y Calculation: If y = 2 ^ x + 3

1 5 If x = 1, then y = 2 ^ 1 + 3 = 2 + 3 = 5

2 7 If x= 2, then y = 2 ^ 2 + 3 = 4 + 3 =7

3 11 If x= 3, then y = 2 ^ 3 + 3 = 11

4 19 If x= 4, then y = 2 ^ 4 + 3 = 16 + 3 =19

The graph would be exponential going upwards with the initial point of (1, 5).

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Recall that the exponentiation in the expression 2^x + 1 must be done before, not after the addition.

When x = 1 we obtain y = 2^1 + 3 = 2 + 3 = 5.

When x = 2 we obtain y = 2^2 + 3 = 4 + 3 = 7.

When x = 3 we obtain y = 2^3 + 3 = 8 + 3 = 11.

When x = 4 we obtain y = 2^4 + 3 = 16 + 3 = 19.

x y

1 5

2 7

3 11

4 19

Looking at the numbers in the y column we see that they increase as we go down the column, and that the increases get progressively larger. In fact if we look carefully we see that each increase is double the one before it, with increases of 2, then 4, then 8.

When we graph these points we find that the graph rises as we go from left to right, and that it rises faster and faster. From our observations on the table we know that the graph in fact that the rise of the graph doubles with each step we take to the right.

The only graph that increases from left to right, getting steeper and steeper with each step, is the graph labeled 'exponential'.

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Self-critique (if necessary):

In this graph there is work that I had to plug in to get the graph. I do not look at this type of math everyday. I could not tell what kind of graph it was so I had to do the work and graph it out on a separate piece of paper to reveal what it was.

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Self-critique Rating:

*********************************************

Question: `q010. If you divide a certain positive number by 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

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Your solution:

It would be equal to the original number because any number divided by one is always going to be equal to the original number.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution: If you divide any number by 1, the result is the same as the original number. Doesn't matter what the original number is, if you divide it by 1, you don't change it.

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Self-critique (if necessary):

------------------------------------------------

Self-critique Rating:

*********************************************

Question: `q011. If you divide a certain positive number by a number greater than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The answer would be less than the original number.

Example:

3 / 2 = 1.5

3 / 6 = .5

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by another number is similar. The bigger the number you divide by, the less you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a number greater than 1, what you get has to be smaller than the original number. Again it doesn't matter what the original number is, as long as it's positive.

Students will often reason from examples. For instance, the following reasoning might be offered:

OK, let's say the original number is 36. Let's divide 36 be a few numbers and see what happens:

36/2 = 18. Now 3 is bigger than 2, and

36 / 3 = 12. The quotient got smaller. Now 4 is bigger than 3, and

36 / 4 = 9. The quotient got smaller again. Let's skip 5 because it doesn't divide evenly into 36.

36 / 6 = 4. Again we divided by a larger number and the quotient was smaller.

I'm convinced.

That is a pretty convincing argument, mainly because it is so consistent with our previous experience. In that sense it's a good argument. It's also useful, giving us a concrete example of how dividing by bigger and bigger numbers gives us smaller and smaller results.

However specific examples, however convincing and however useful, don't actually prove anything. The argument given at the beginning of this solution is general, and applies to all positive numbers, not just the specific positive number chosen here.

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Self-critique (if necessary):

------------------------------------------------

Self-critique Rating:

*********************************************

Question: `q012. If you divide a certain positive number by a positive number less than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The answer is the result would be greater than the original number.

Example:

12 / .5 = 24

12/.3 = 40

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by some other number is similar. The bigger the number you divide by, the less you get. The smaller the number you divide by, the more you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a positive number less than 1, what you get has to be larger than the original number. Again it doesn't matter what the original number is, as long as it's positive.

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Self-critique (if necessary):

------------------------------------------------

Self-critique Rating:

*********************************************

Question: `q013. Students often get the basic answers to nearly all, or even all these questions, correct. Your instructor has however never seen anyone who addressed all the subtleties in the given solutions in their self-critiques, and it is very common for a student to have given no self-critiques. It is very likely that there is something in the given solutions that is not expressed in your solution.

This doesn't mean that you did a bad job. If you got most of the 'answers' right, you did fine.

However, in order to better understand the process, you are asked here to go back and find something in one of the given solutions that you did not address in your solution, and insert a self-critique. You should choose something that isn't trivial to you--something you're not 100% sure you understand.

If you can't find anything, you can indicate this below, and the instructor will point out something and request a response (the instructor will select something reasonable, but will then expect a very good and complete response). However it will probably be less work for you if you find something yourself.

Your response should be inserted at the appropriate place in this document, and should be indicated by preceding it with ####.

As an answer to this question, include a copy of whatever you inserted above, or an indication that you can't find anything.

your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

________________________________________

Submit a copy of this document using the Submit Work Form at http://vhcc2.vhcc.edu/dsmith/submit_work.htm. The form has instructions but read the following:

• You will be asked to give your work a title. You may use any title you wish; if you aren't sure what you want to call it, just call it 'First Two Questions' or something of that nature. The title you choose is the title under which your work will be posted after the instructor has reviewed it.

• You will simply copy and paste everything that precedes this paragraph, including your answers, Confidence Ratings, self-critiques, etc., into a box in the form, and click Submit.

• Your work will then be posted by the end of the following day, and often by the end of the day on which you submit it, at your personal access site. You received instructions for accessing this site with your access code.

It is suggested that you bookmark the Submit Work Form now, but if you don't you will be reminded later.

________________________________________

When you have submitted this document, you will have complete Step 3 of the 8-step Orientation. Your next step will be to return tohttp://vhcc2.vhcc.edu/dsmith/geninfo/startup_and_orientation.htm and continue on to Step 4.

"

Self-critique (if necessary):

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Self-critique rating:

________________________________________

&#Your work looks good. Let me know if you have any questions. &#

#$&*

#(*!

course Mth 173

1/11 10

Introductory Question-Answer (qa) Sequence________________________________________

There are 12 questions in this document, along with some instructions.

Copy this document into a word processor or text editor.

Answer the first two questions below, inserting your answers, Confidence Ratings and self-critiques as explained. You will then read the instructions that precede the remaining ten questions and answer those questions in a similar manner. Finally you will submit your work using a web form, according to instructions at the end of the document.

You will probably find that you can answer many of these questions without writing anything down. On those problems where you cannot arrive at an answer 'in your head', is recommended that you work out your solutions on paper. It is often helpful to sketch, doodle, jot down ideas, do calculations, organize and test ideas on paper.

When appropriate, you will later be encouraged to use a calculator to do any arithmetic you cannot do mentally. However the calculator is not appropriate to the questions that appear on this document. Put the calculator aside and think through these questions.

________________________________________

Here is some additional information on the process and how it will fit into your course:

One of the predominant features of your course is the question-answer format for submitting work.

• In most courses you will encounter sequenced questions of this nature, designed to build your understanding by engaging you in the process of answering and self-critiquing your answers on a number of questions.

• In all courses you will submit assigned problems using this format.

As with the first couple of questions, the questions in this document can be answered with just a knowledge of basic high-school mathematics.

Sometimes the given solutions are more subtle than you might expect, and you will probably find that many of your answers, while good and correct, do not completely match the given solution. This is intentional, the goal being to get you used to the idea and the benefits of the self-critique process.

• Don't worry if you have trouble with a few of the questions, or if your explanations don't quite match those in the given solutions--most students begin their course a little rusty.

• Be sure to do your best to understand all the questions and the given solutions--it's this effort that makes the process beneficial to you.

The process is fairly simple, and you'll be using it again and again.

• The process will soon be very familiar to you, if it isn't already.

• Work through the instructions given here and within the questions, and do your best.

• If you miss something in the process (as most students are bound to do the first time through), your instructor will point it out to you, and there will be ample opportunity to get everything straight.

________________________________________

Your basic instructions follow. Rather than giving you the instructions at the beginning of the document, you were given a couple of questions to serve as a point of reference, and should now be better prepared to understand the instructions:

1. Answer each question, then look at the given solution:

It is expected that you will answer each question before looking at the given solution. There is no grade penalty for looking ahead, but if you do you:

• may be bypassing an opportunity to engage yourself in the solution process

• run the risk of deluding yourself about what you understand

• are likely to learn much less and

• are not as likely to do well in the course.

However your instructor understands the tradeoffs involved in being a student, and makes no judgement about how you should use this material. As long as you use it to your best advantage and succeed on tests and other course activities, you will get a good result from this course.

2. If you can't readily work it out in your head, use pencil and paper, and keep a record of your work.

You aren't expected to work out your solutions by staring at a computer screen, though you will likely find many questions and problems easy enough to do 'in your head'. However on more challenging problems, it's easier to work things out using a handwritten document than a computer-created or word-processed document.

• You should in general work out your answers to non-obvious problems on paper, jotting down sketches, diagrams and notes as you go, in such a way that you can make sense of it later. This will help you focus your work and maintain your train of thought, and will be quite valuable for periodic review. It is recommended (and may in some courses be required) that you dedicate a notebook to this course, and at least sketch out your work in the notebook.

3. There's no need for special formatting or graphs:

• Don't use special characters in your responses (e.g., characters like  that don't appear on your keyboard). The characters on your keyboard suffice to answer all these questions. If you use special characters they won't come through the form you use to submit your work, and if you use too many such characters your instructor might not be able to tell what you are saying.

• Don't try to make graphs in your document. Sketch your graphs by hand, then if necessary describe them in words (that probably won't be necessary in the present exercise; more about that later). Graphs won't come through when you submit your work. You can make a graph without understanding it, but you can't give a good description of your graph without understanding it. Your instructor doesn't need to see your graphs; he needs to see your descriptions of your graphs. The present exercise doesn't require extensive descriptions of graphs; it they apply to your course, you will see more about describing your graphs soon.

• When you submit this document (per instructions at the end), it will come to the instructor in pure text format. Any formatting you have done will not be seen by the instructor, special characters will not appear in what the instructor sees, and graphs won't come through. So don't do any fancy formatting for the instructor. You will of course want to save your original copy, and you are welcome to add formatting for your own purposes.

4. Expect to see your work posted by the instructor in a timely manner:

After submitting your work, per instructions at the end of this document, you should expect to see your work posted, along with instructor commentary, at your access page. It should be posted by the evening of the day after you submit it, and may well be posted the evening of the same day.

You have probably submitted your answers the first two questions in a preceding task. You are welcome to answer them again, but if you have already submitted them you may go ahead and skip to the third question.

*********************************************

Question: `q001. If you are earning money at the rate of 8 dollars / hour and work for 4 hours, how much money do you make during this time? Answer in such a way as to explain your reasoning as fully as possible. A solution to this problem appears several lines below, but enter your own solution before you look at the given solution.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

What I would do to get the solution is to multiply the rate of money made ( 8 dollars ) by how many hours worked ( 4 hours ). The answer would be 32 dollars an hour.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: 8 dollars / hour means '8 dollars per hour', indicating that for every hour you work you earn 8 dollars. If you work for 4 hours, then if you earn 8 dollars for every one of those hours you earn 4 * 8 dollars = 32 dollars.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

ok

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

*********************************************

Question: `q002. If you work 12 hours and earn $168, then at what rate, in dollars / hour, were you making money?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

To find the solution I would divide the money earned ( 168 dollars ) by the hours worked. That would make the answer 14 dollars an hour.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: $168 earned in 12 hours implies that $168 / 12 = $14 were made per hour, so the rate is $14 / hour.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

ok

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

________________________________________

________________________________________

Here are the remaining ten questions:

*********************************************

Question: `q003. If you are earning 8 dollars / hour, how long will it take you to earn $72? The answer may well be obvious, but explain as best you can how you reasoned out your result.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

Divide $72 by 8 dollars so the hours worked will show. The answer is 9 hours worked.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

.............................................

Given Solution: Many students simply know, at the level of common sense, that if we divide $72 by $8 / hour we get 9 hours, so 9 hours are required.

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Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

ok

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

*********************************************

Question: `q004. Calculate (8 + 3) * 5 and 8 + 3 * 5, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

Order of operations.

(8 + 3) * 5

First parenthesis (8 + 3) = 11

Second times 11 * 5 = 55

Solution = 55

8 + 3 * 5

First times 3 * 5 = 15

Second add 8 + 15 = 23

The reason the answers are different are because the way they are wrote. One has parenthesis the other does not.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: (8 + 3) * 5 and 8 + 3 * 5

To evaluate (8 + 3) * 5, you will first do the calculation in parentheses. 8 + 3 = 11, so

(8 + 3) * 5 = 11 * 5 = 55.

To evaluate 8 + 3 * 5 you have to decide which operation to do first, 8 + 3 or 3 * 5. You should be familiar with the order of operations, which tells you that multiplication precedes addition. The first calculation to do is therefore 3 * 5, which is equal to 15. Thus

8 + 3 * 5 = 8 + 15 = 23

The results are different because the grouping in the first expression dictates that the addition be done first.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

• If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

• If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

ok

------------------------------------------------

Self-critique Rating: 3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

In subsequent problems the detailed instructions that accompanied the first four problems are missing. We assume you will know to follow the same instructions in answering the remaining questions.

*********************************************

Question: `q005. Calculate (2^4) * 3 and 2^(4 * 3), indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. Note that the symbol '^' indicates raising to a power. For example, 4^3 means 4 raised to the third power, which is the same as 4 * 4 * 4 = 64.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Order of operations

(2^4) * 3

First parenthesis 2 to the fourth power = 16

Second times 16 * 3 = 48

2^(4 * 3)

First parenthesis (4*3) = 12

Second powers 2^12 = 4096

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

.............................................

Given Solution: 3

To evaluate (2^4) * 3 we first evaluate the grouped expression 2^4, which is the fourth power of 2, equal to 2 * 2 * 2 * 2 = 16. So we have

(2^4) * 3 = 16 * 3 = 48.

To evaluate 2^(4 * 3) we first do the operation inside the parentheses, obtaining 4 * 3 = 12. We therefore get

2^(4 * 3) = 2^12 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 4096.

It is easy to multiply by 2, and the powers of 2 are important, so it's appropriate to have asked you to do this problem without using a calculator. Had the exponent been much higher, or had the calculation been, say, 3^12, the calculation would have become tedious and error-prone, and the calculator would have been recommended.

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Self-critique (if necessary):

ok

*********************************************

Question: `q006. Calculate 3 * 5 - 4 * 3 ^ 2 and 3 * 5 - (4 * 3)^2 according to the standard order of operations, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

3 * 5 – 4 * 3 ^ 2

First powers 3 ^ 2 = 9

Second multiply 3 * 5 = 15 and 4 * 9 = 36

Third subtract 15 – 36 = -21

3 * 5 – (4 * 3) ^ 2

First parenthesis (4 * 3) = 12

Second powers 12 ^ 2 = 144

Third multiply 3 * 5 = 15

Fourth subtract 15 – 144 = 129

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

To calculate 3 * 5 - 4 * 3 ^ 2, the first operation is the exponentiation operation ^.

• The two numbers involved in the exponentiation are 3 and 2; the 4 is 'attached' to the 3 by multiplication, and this multiplication can't be done until the exponentiation has been performed.

• The exponentiation operation is therefore 3^2 = 9, and the expression becomes 3 * 5 - 4 * 9.

Evaluating this expression, the multiplications 3 * 5 and 4 * 9 must be performed before the subtraction. 3 * 5 = 15 and 4 * 9 = 36 so we now have

3 * 5 - 4 * 3 ^ 2 = 3 * 5 - 4 * 9 = 15 - 36 = -21.

To calculate 3 * 5 - (4 * 3)^2 we first do the operation in parentheses, obtaining 4 * 3 = 12. Then we apply the exponentiation to get 12 ^2 = 144. Finally we multiply 3 * 5 to get 15. Putting this all together we get

3 * 5 - (4 * 3)^2 =

3 * 5 - 12^2 =

3 * 5 - 144 =

15 - 144 =

-129.

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Self-critique (if necessary):

ok

------------------------------------------------

Self-critique Rating: 3

In the next three problems, the graphs will be of one of the basic shapes listed below. You will be asked to construct graphs for three simple functions, and determine which of the depicted graphs each of your graphs most closely resembles. At this point you won't be expected to know these terms or these graph shapes; if at some point in your course you are expected to know these things, they will be presented at that point.

Linear:

Quadratic or parabolic:

Exponential:

Odd power:

Fractional positive power:

Even negative power:

partial graph of polynomial of degree 3

more extensive graph of polynomial of degree 3

*********************************************

Question: `q007. Let y = 2 x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

-2

-1

0

1

2

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

y = 2 x + 3.

For -2 = 2(-2) +3

-4 + 3 = -1

For -1 = 2 (-1) + 3

-2 + 3 = 1

For 0 = 2(0) + 3

0 + 3 = 3

For 1 = 2(1) + 3

2 + 3 = 5

For 2 = 2(2) + 3

4 + 3 = 7

The graph is a linear graph that goes through the points of (-2,-1), (-1,1), (0, 3), (1, 5), (2,7). It crosses the y axis at 3.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Two slightly different explanations are give below, one by a student and one by the instructor. Neither format is inherently better than the other.

GOOD SOLUTION BY STUDENT:

First we need to complete the table. I have added a column to the right of the table to show the calculation of “y” when we us the “x” values as given.

x y Calculation: If y = 2x + 3

-2 -1 If x = -2, then y = 2(-2)+3 = -4+3 = -1

-1 1 If x= -1, then y = 2(-1)+3 = -2+3 = 1

0 3 If x= 0, then y = 2(0)+3 = 0+3 = 3

1 5 If x= 1, then y = 2(1)+3 = 2+3 = 5

2 7 If x= 2, then y = 2(2)+3 = 4+3 = 7

Once an answer has been determined, the “y” value can be filled in. Now we have both the “x” and “y” values and we can begin our graph. The charted values continue on a straight line representing a linear function as shown above.

INSTRUCTOR'S SOLUTION:

We easily evaluate the expression:

• When x = -2, we get y = 2 x + 3 = 2 * (-2) + 3 = -4 + 3 = -1.

• When x = -1, we get y = 2 x + 3 = 2 * (-1) + 3 = -2 + 3 = 1.

• When x = 0, we get y = 2 x + 3 = 2 * (0) + 3 = 0 + 3 = 3.

• When x = 1, we get y = 2 x + 3 = 2 * (1) + 3 = 2 + 3 = 5.

• When x = 2, we get y = 2 x + 3 = 2 * (2) + 3 = 4 + 3 = 7.

Filling in the table we have

x y

-2 -1

-1 1

0 3

1 5

2 7

When we graph these points we find that they lie along a straight line.

Only one of the depicted graphs consists of a straight line, and we conclude that the appropriate graph is the one labeled 'linear'.

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Self-critique (if necessary):

------------------------------------------------

Self-critique Rating:

*********************************************

Question: `q008. Let y = x^2 + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

-2

-1

0

1

2

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

y = x^2 + 3

x y Calculation: If y = x^2 + 3

-2 7 If x = -2, then y = -2^2 + 3 = 4 + 3 = 7

-1 4 If x= -1, then y = -1^2 + 3 = 1 + 3 = 4

0 3 If x= 0, then y = 0^2 + 3 = 0+3 = 3

1 4 If x= 1, then y = 1^2 + 3 = 4

2 7 If x= 2, then y = 2^2 + 3 = 7

The graph would be a parabola with the vertex of (0, 3) and facing up.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

Evaluating y = x^2 + 3 at the five points:

• If x = -2 then we obtain y = x^2 + 3 = (-2)^2 + 3 = 4 + 3 = 7.

• If x = -1 then we obtain y = x^2 + 3 = (-1)^2 + 3 = ` + 3 = 4.

• If x = 0 then we obtain y = x^2 + 3 = (0)^2 + 3 = 0 + 3 = 3.

• If x = 1 then we obtain y = x^2 + 3 = (1)^2 + 3 = 1 + 3 = 4.

• If x = 2 then we obtain y = x^2 + 3 = (2)^2 + 3 = 4 + 3 = 7.

The table becomes

x y

-2 7

-1 4

0 3

1 4

2 7

We note that there is a symmetry to the y values. The lowest y value is 3, and whether we move up or down the y column from the value 3, we find the same numbers (i.e., if we move 1 space up from the value 3 the y value is 4, and if we move one space down we again encounter 4; if we move two spaces in either direction from the value 3, we find the value 7).

A graph of y vs. x has its lowest point at (0, 3).

If we move from this point, 1 unit to the right our graph rises 1 unit, to (1, 4), and if we move 1 unit to the left of our 'low point' the graph rises 1 unit, to (-1, 4).

If we move 2 units to the right or the left from our 'low point', the graph rises 4 units, to (2, 7) on the right, and to (-2, 7) on the left.

Thus as we move from our 'low point' the graph rises up, becoming increasingly steep, and the behavior is the same whether we move to the left or right of our 'low point'. This reflects the symmetry we observed in the table. So our graph will have a right-left symmetry.

Two of the depicted graphs curve upward away from the 'low point'. One is the graph labeled 'quadratic or parabolic'. The other is the graph labeled 'partial graph of degree 3 polynomial'.

If we look closely at these graphs, we find that only the first has the right-left symmetry, so the appropriate graph is the 'quadratic or parabolic' graph.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `q009. Let y = 2 ^ x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

• Evaluate y for x = 1. What is your result? In your solution explain the steps you took to get this result.

• Evaluate y for x values 2, 3 and 4. Write out a copy of the table below. In your solution give the y values you obtained in your table.

x y

1

2

3

4

• Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

• In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

x y Calculation: If y = 2 ^ x + 3

1 5 If x = 1, then y = 2 ^ 1 + 3 = 2 + 3 = 5

2 7 If x= 2, then y = 2 ^ 2 + 3 = 4 + 3 =7

3 11 If x= 3, then y = 2 ^ 3 + 3 = 11

4 19 If x= 4, then y = 2 ^ 4 + 3 = 16 + 3 =19

The graph would be exponential going upwards with the initial point of (1, 5).

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

Recall that the exponentiation in the expression 2^x + 1 must be done before, not after the addition.

When x = 1 we obtain y = 2^1 + 3 = 2 + 3 = 5.

When x = 2 we obtain y = 2^2 + 3 = 4 + 3 = 7.

When x = 3 we obtain y = 2^3 + 3 = 8 + 3 = 11.

When x = 4 we obtain y = 2^4 + 3 = 16 + 3 = 19.

x y

1 5

2 7

3 11

4 19

Looking at the numbers in the y column we see that they increase as we go down the column, and that the increases get progressively larger. In fact if we look carefully we see that each increase is double the one before it, with increases of 2, then 4, then 8.

When we graph these points we find that the graph rises as we go from left to right, and that it rises faster and faster. From our observations on the table we know that the graph in fact that the rise of the graph doubles with each step we take to the right.

The only graph that increases from left to right, getting steeper and steeper with each step, is the graph labeled 'exponential'.

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Self-critique (if necessary):

In this graph there is work that I had to plug in to get the graph. I do not look at this type of math everyday. I could not tell what kind of graph it was so I had to do the work and graph it out on a separate piece of paper to reveal what it was.

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Self-critique Rating:

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Question: `q010. If you divide a certain positive number by 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

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Your solution:

It would be equal to the original number because any number divided by one is always going to be equal to the original number.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: If you divide any number by 1, the result is the same as the original number. Doesn't matter what the original number is, if you divide it by 1, you don't change it.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `q011. If you divide a certain positive number by a number greater than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

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Your solution:

The answer would be less than the original number.

Example:

3 / 2 = 1.5

3 / 6 = .5

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by another number is similar. The bigger the number you divide by, the less you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a number greater than 1, what you get has to be smaller than the original number. Again it doesn't matter what the original number is, as long as it's positive.

Students will often reason from examples. For instance, the following reasoning might be offered:

OK, let's say the original number is 36. Let's divide 36 be a few numbers and see what happens:

36/2 = 18. Now 3 is bigger than 2, and

36 / 3 = 12. The quotient got smaller. Now 4 is bigger than 3, and

36 / 4 = 9. The quotient got smaller again. Let's skip 5 because it doesn't divide evenly into 36.

36 / 6 = 4. Again we divided by a larger number and the quotient was smaller.

I'm convinced.

That is a pretty convincing argument, mainly because it is so consistent with our previous experience. In that sense it's a good argument. It's also useful, giving us a concrete example of how dividing by bigger and bigger numbers gives us smaller and smaller results.

However specific examples, however convincing and however useful, don't actually prove anything. The argument given at the beginning of this solution is general, and applies to all positive numbers, not just the specific positive number chosen here.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `q012. If you divide a certain positive number by a positive number less than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

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Your solution:

The answer is the result would be greater than the original number.

Example:

12 / .5 = 24

12/.3 = 40

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by some other number is similar. The bigger the number you divide by, the less you get. The smaller the number you divide by, the more you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a positive number less than 1, what you get has to be larger than the original number. Again it doesn't matter what the original number is, as long as it's positive.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: `q013. Students often get the basic answers to nearly all, or even all these questions, correct. Your instructor has however never seen anyone who addressed all the subtleties in the given solutions in their self-critiques, and it is very common for a student to have given no self-critiques. It is very likely that there is something in the given solutions that is not expressed in your solution.

This doesn't mean that you did a bad job. If you got most of the 'answers' right, you did fine.

However, in order to better understand the process, you are asked here to go back and find something in one of the given solutions that you did not address in your solution, and insert a self-critique. You should choose something that isn't trivial to you--something you're not 100% sure you understand.

If you can't find anything, you can indicate this below, and the instructor will point out something and request a response (the instructor will select something reasonable, but will then expect a very good and complete response). However it will probably be less work for you if you find something yourself.

Your response should be inserted at the appropriate place in this document, and should be indicated by preceding it with ####.

As an answer to this question, include a copy of whatever you inserted above, or an indication that you can't find anything.

your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

________________________________________

Submit a copy of this document using the Submit Work Form at http://vhcc2.vhcc.edu/dsmith/submit_work.htm. The form has instructions but read the following:

• You will be asked to give your work a title. You may use any title you wish; if you aren't sure what you want to call it, just call it 'First Two Questions' or something of that nature. The title you choose is the title under which your work will be posted after the instructor has reviewed it.

• You will simply copy and paste everything that precedes this paragraph, including your answers, Confidence Ratings, self-critiques, etc., into a box in the form, and click Submit.

• Your work will then be posted by the end of the following day, and often by the end of the day on which you submit it, at your personal access site. You received instructions for accessing this site with your access code.

It is suggested that you bookmark the Submit Work Form now, but if you don't you will be reminded later.

________________________________________

When you have submitted this document, you will have complete Step 3 of the 8-step Orientation. Your next step will be to return tohttp://vhcc2.vhcc.edu/dsmith/geninfo/startup_and_orientation.htm and continue on to Step 4.

"

@& Your explanations are good.

You will need to include condfidence ratings in the future.*@

#(*!#(*!

course Mth 173

1/11/2011 11:03pm

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

.

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

003. Misc: Surface Area, Pythagorean Theorem, Density

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Question: `q001. What is surface area of a rectangular solid whose dimensions are 3 meters by 4 meters by 6 meters?

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Your solution:

The surface area of the rectangular solid is 108m^2.

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`aA rectangular solid has six faces (top, bottom, front, back, left side, right side if you're facing it). The pairs top and bottom, right and left sides, and front-back have identical areas. This solid therefore has two faces with each of the following dimensions: 3 m by 4 m, 3 m by 6 m and 4 m by 6 m, areas 12 m^2, 18 m^2 and 24 m^2. Total area is 2 * 12 m^2 + 2 * 18 m^2 + 2 * 24 m^2 = 108 m^2.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q002. What is the surface area of the curved sides of a cylinder whose radius is five meters and whose altitude is 12 meters? If the cylinder is closed what is its total surface area?

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Your solution:

The surface area of the sides of the cylinder is 376.99m^2. The total surface area of the cylinder is 534.1m^2.

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

The circumference of this cylinder is 2 pi r = 2 pi * 5 m = 10 pi m. If the cylinder was cut by a straight line running up its curved face then unrolled it would form a rectangle whose length and width would be the altitude and the circumference. The area of the curved side is therefore

A = circumference * altitude = 10 pi m * 12 m = 120 pi m^2.

If the cylinder is closed then it has a top and a bottom, each a circle of radius 5 m with resulting area A = pi r^2 = pi * (5 m)^2 = 25 pi m^2. The total area would then be

total area = area of sides + 2 * area of base = 120 pi m^2 + 2 * 25 pi m^2 = 170 pi m^2.

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Self-critique (if necessary):

I only used the actual value of the solution in my answer instead of the process to get my answer. I understand the arithmetic used to get the answer because I used the same arithmetic and wrote the problem in a notebook.

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Self-critique Rating: 2

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Question: `q003. What is surface area of a sphere of diameter three cm?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The surface area of the sphere is 28.27 cm^2. 4*pi*(1.5)^2= 4*pi*2.25 = 9*pi

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`aThe surface area of a sphere of radius r is A = 4 pi r^2. This sphere has radius 3 cm / 2, and therefore has surface area

A = 4 pi r^2 = 4 pi * (3/2 cm)^2 = 9 pi cm^2.

NOTE TO STUDENT:

While your work on most problems has been good, you left this problem blank and didn't self-critique.

You should self-critique here.

• For example you should acknowledge having made note of the formula for the surface area of the sphere, which I expect you didn't know before.

I expect from your previous answers that you are very capable of applying the formula once you have it, and based on this history you probably wouldn't need to self-critique that aspect of the process.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q004. What is hypotenuse of a right triangle whose legs are 5 meters and 9 meters?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The hypotenuse is 10.3 meters^2

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`aThe Pythagorean Theorem says that the hypotenuse c of a right triangle with legs a and b satisfies the equation c^2 = a^2 + b^2. So, since all lengths are positive, we know that

c = sqrt(a^2 + b^2) = sqrt( (5 m)^2 + (9 m)^2 ) = sqrt( 25 m^2 + 81 m^2) = sqrt( 106 m^2 ) = 10.3 m, approx..

Note that this is not what we would get if we made the common error of assuming that sqrt(a^2 + b^2) = a + b; this would tell us that the hypotenuse is 14 m, which is emphatically not so. There is no justification whatsoever for applying a distributive law (like x * ( y + z) = x * y + x * z ) to the square root operator.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q005. If the hypotenuse of a right triangle has length 6 meters and one of its legs has length 4 meters what is the length of the other leg?

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Your solution:

The length of the other leg is 4.47m, or sqrt(20)

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`aIf c is the hypotenuse and a and b the legs, we know by the Pythagorean Theorem that c^2 = a^2 + b^2, so that a^2 = c^2 - b^2. Knowing the hypotenuse c = 6 m and the side b = 4 m we therefore find the unknown leg:

a = sqrt( c^2 - b^2) = sqrt( (6 m)^2 - (4 m)^2 ) = sqrt(36 m^2 - 16 m^2) = sqrt(20 m^2) = sqrt(20) * sqrt(m^2) = 2 sqrt(5) m,

or approximately 4.4 m.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q006. If a rectangular solid made of a uniform, homogeneous material has dimensions 4 cm by 7 cm by 12 cm and if its mass is 700 grams then what is its density in grams per cubic cm?

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Your solution:

The density of the rectangular solid is 2.08 g/cm^3

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`aThe volume of this solid is 4 cm * 7 cm * 12 cm = 336 cm^3.

Its density in grams per cm^3 is the number of grams in each cm^3. We find this quantity by dividing the number of grams by the number of cm^3. We find that

• density = 700 grams / (336 cm^3) = 2.06 grams / cm^3.

Note that the solid was said to be uniform and homogeneous, meaning that it's all made of the same material, which is uniformly distributed. So each cm^3 does indeed have a mass of 2.06 grams.

• Had we not known that the material was uniform and homogeneous we could have said that the average density is 2.06 grams / cm^3, but not that the density is 2.06 grams / cm^3 (for example the object could be made of two separate substances, one with density less than 2.06 grams / cm^3 and the other with density greater than 2.06 g / cm^3, in appropriate proportions; neither substance would have density 2.06 g / cm^3, but the average density could be 2.06 g / cm^3).

NOTE TO STUDENT: (in this note the instructor attempts to clarify the idea of 'demonstrating what you do and do not understand about the statement of the problem' and 'giving a phrase-by-phrase analysis of the given solution')

You did not respond to the question and did not self-critique.

You would be expected to address the question, stating what you do and do not understand.

• For example you should understand what a rectangular solid with dimensions 4 cm by 7 cm by 12 cm is, and how to find its volume and surface area. You might not know what to do with this information (for example you might well not understand that it's the volume and not the surface area that's related to density), but from previous work you should understand this much, and should at least mention something along the lines of 'well, I do know that I can find the volume and/or surface area of that solid' in a partial solution.

• The word 'density' is clearly very important. Even if you don't know what density is, you could note from the statement of the problem that its units here are said to be 'grams per cubic centimeter'.

Having noted these things, you will be much better prepared to understand the information in the given solution.

Then you need to address the information in the given solution. A 'phrase-by-phrase' analysis is generally very beneficial:

• I expect you understand the first statement from previous knowledge (you should have this understanding from prerequisite courses, and if not you encountered it in the preceding 'volumes' exercise): 'The volume of this solid is 4 cm * 7 cm * 12 cm = 336 cm^3.' It would of course be appropriate to ask a question here if necessary.

• It is likely that, as is the case with many students, the concept of density is not that familiar to you. However if this wasn't addressed specifically in prerequisite courses, those courses would be expected to prepare you to understand this concept. The statement 'Its density in grams per cm^3 is the number of grams in each cm^3.' serves as a definition of density. In your self-critique you should have addressed what what this phrase means to you, and what you do or do not understand about it

• The next phrase is 'We find this quantity by dividing the number of grams by the number of cm^3.' You would be expected to understand that this phrase is related to the preceding, and as best you can to address the connection. At this point many students would need to ask a question, and it would be perfectly appropriate to do so (or to have done so regarding previous statements).

• The subsequent phrase 'density = 700 grams / (336 cm^3) = 2.06 grams / cm^3' is an illustration of the ideas and definitions in the preceding statements. A reasonable self-critique would demonstrate your attempt to understand this statement and its connection to the preceding. Once again questions would also be appropriate and welcome.

• The above addresses sufficient information to solve the problem. If you get to this point, you're probably doing OK and you wouldn't necessarily be expected to address the rest of the given solution, which expands on the finer details of the problem and provides additional information. The basic prerequisite courses should have prepared you to understand the information, but students entering Liberal Arts Mathematics, College Algebra and even Precalculus or Applied Calculus (or Physics 121-122) courses probably don't need to address anything beyond the basic solution at this point. Though Precalculus and Applied Calculus students could benefit from doing so, and if time permits would certainly be encouraged to do so, time is also a factor and it would be understandable if these students chose to move on.

• Students entering the Mth 173-4 sequence or the Phy 201-202 or 231-232 sequence would be expected to either completely understand all the details of the given solution, or address them in your self-critique.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q007. What is the mass of a sphere of radius 4 meters if its average density is 3,000 kg/cubic meter?

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Your solution:

The mass of the sphere is 804240 kg.

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`aA average density of 3000 kg / cubic meter implies that, at least on the average, every cubic meter has a mass of 3000 kg. So to find the mass of the sphere we multiply the number of cubic meters by 3000 kg.

The volume of a sphere of radius 4 meters is 4/3 pi r^3 = 4/3 * pi (4m)^3 = 256/3 * pi m^3. So the mass of this sphere is

mass = density * volume = 256 / 3 * pi m^3 * 3000 kg / m^3 = 256,000 * pi kg.

This result can be approximated to an appropriate number of significant figures.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q008. If we build an object out of two pieces of material, one having a volume of 6 cm^3 at a density of 4 grams per cm^3 and another with a volume of 10 cm^3 at a density of 2 grams per cm^3 then what is the average density of this object?

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Your solution:

6cm^3*4g/cm^3= 24g, 10cm^3*2g/cm^3= 20g, 24+20= 44g. The combined volume is 6cm^3+10cm^3= 16cm^3. The average density is 44g/16cm^3= 2.75g/cm^3.

confidence rating #$&*: 3

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Given Solution:

`aThe first piece has a mass of 4 grams / cm^3 * 6 cm^3 = 24 grams. The second has a mass of 2 grams / cm^3 * 10 cm^3 = 20 grams. So the total mass is 24 grams + 20 grams = 44 grams.

The average density of this object is

average density = total mass / total volume = (24 grams + 20 grams) / (6 cm^3 + 10 cm^3) = 44 grams / (16 cm^3) = 2.75 grams / cm^3.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q009. In a large box of dimension 2 meters by 3 meters by 5 meters we place 27 cubic meters of sand whose density is 2100 kg/cubic meter, surrounding a total of three cubic meters of cannon balls whose density is 8,000 kg per cubic meter. What is the average density of the material in the box?

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Your solution:

The total mass is found by finding the mass of the sand (2100kg/cm^3*27cm^3 = 56700kg) and finding the mass of the cannon balls (8000kg/cm^3*3cm^3 = 24,000kg) and adding the two massed together (56700kg + 24,000kg = 80,700kg). Then, the total mass is divided by the total volume to get the average density, 80,700kg/30cm^3 = 2,690kg/cm^3.

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`aWe find the average density from the total mass and the total volume. The mass of the sand is 27 m^3 * 2100 kg / m^3 = 56,700 kg. The mass of the cannonballs is 3 m^3 * 8,000 kg / m^3 = 24,000 kg.

The average density is therefore

average density = total mass / total volume = (56,700 kg + 24,000 kg) / (27 m^3 + 3 m^3) = 80,700 kg / (30 m^3) = 2,700 kg / m^3, approx..

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q010. How many cubic meters of oil are there in an oil slick which covers 1,700,000 square meters (between 1/2 and 1 square mile) to an average depth of .015 meters? If the density of the oil is 860 kg/cubic meter the what is the mass of the oil slick?

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Your solution:

The volume of the oil slick (a cylinder) is 25,500m^3. The density of the oil slick is 860kg/m^3, so the mass of the oil slick is 21,930,000 kg.

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`aThe volume of the slick is V = A * h, where A is the area of the slick and h the thickness. This is the same principle used to find the volume of a cylinder or a rectangular solid. We see that the volume is

V = A * h = 1,700,000 m^2 * .015 m = 25,500 m^3.

The mass of the slick is therefore

mass = density * volume = 860 kg / m^3 * 25,500 m^3 = 21 930 000 kg.

This result should be rounded according to the number of significant figures in the given information.

STUDENT QUESTION

I didn’t round to the most significant figure. ???? How important is this?

INSTRUCTOR RESPONSE

It will be important.

This document is preliminary; the issue of significant figures will be addressed more specifically as we move into the course.

Right now I just want you to be aware of the general idea.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q011. Part 1 Summary Question 1: How do we find the surface area of a cylinder?

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Your solution:

The surface area of a cylinder is its radius multiplied by pi then doubled, then added to the product of its radius multiplied by its height and multiplied by pi and doubled.

confidence rating #$&*:

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Given Solution:

`aThe curved surface of the cylinder can be 'unrolled' to form a rectangle whose dimensions are equal to the circumference and the altitude of the cylinder, so the curved surface has volume

Acurved = circumference * altitude = 2 pi r * h, where r is the radius and h the altitude.

The top and bottom of the cylinder are both circles of radius r, each with resulting area pi r^2.

{]The total surface area is therefore

Acylinder = 2 pi r h + 2 pi r^2.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q012. Part 1 Summary Question 2: What is the formula for the surface area of a sphere?

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Your solution:

The surface area of a sphere is found with the formula 4*pi*r^2

confidence rating #$&*: 3

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Given Solution:

`aThe surface area of a sphere is

A = 4 pi r^2,

where r is the radius of the sphere.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q013. Part 1 Summary Question 3: What is the meaning of the term 'density'.

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Your solution:

Density is mass with respect to volume. D=m/v

confidence rating #$&*: OK

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Given Solution:

`aThe average density of an object is its mass per unit of volume, calculated by dividing its total mass by its total volume. If the object is uniform and homogeneous then its density is constant and we can speak of its 'density' as opposed to its 'average density'

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q014. Part 1 Summary Question 4: If we know average density and mass, how can we find volume?

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Your solution:

If the average density and mass is known, volume is found by dividing the mass by the density.

confidence rating #$&*: 3

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Given Solution:

Since mass = ave density * volume, it follows by simple algebra that volume = mass / ave density.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q015. Part 1 Explain how you have organized your knowledge of the principles illustrated by the exercises in this assignment.

I have written all of the formulas in an assigned notebook for this class, to be organized.

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&#Your work looks good. Let me know if you have any questions. &#