Orientation Step 7

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course Mth 271

Task: `q001. If you were in a course that meets in a classroom you would be attending regularly. In an asychronous distance course, while you have the course materials and access to a great deal of instruction, you do not have the benefit of regular meetings, and it can be difficult to find the time to work on the course.

You cannot allow this course to become something you need to 'find time' to do. You need to schedule a regular time to work on this course, and you need to schedule a sufficient number of hours to do this work.

Here's the arithmetic of being a college student:

• The generally accepted standard is that at the minimum, it's a full-time job to be a full-time student.

• A full-time job for 16 weeks translates to a total of 640 hours, over the course of a semester, devoted to a 15-credit courseload.

• Approximately 240-300 of them in class and/or lab and the remainder devoted to preparation and study outside the classroom.

• This comes to something over 40 hours per credit-hour. That's 120 hours for a 3-credit class, 160 hours for a 4-credit class and 200 hours for a 5-credit class, spread over 16 weeks.

There is of course a wide degree of variation in the time actually required of an individual student:

• Some courses require less time than others.

• Students vary in the knowledge they bring from prerequisite courses.

• Students learn at varying paces, some more quickly and others more slowly.

• Study habits and efficiency of time use vary widely among students.

So not everyone requires all those hours, but some will require more.

• You should begin this course with the assumption that you will require about the number of hours specified above.

Though there are exceptions both ways, most people manage to establish a regular schedule are successful in these courses, and most people who fail to establish a regular schedule are not successful.

Please explain in your own words why it is important to establish a schedule for this course, and to put aside the required number of hours.

**** Your response (insert your response beginning in the next line; the next line is blank and doesn't include the #$... prompt):

It is easy to get behind and it is very important to stay on schedule completing tasks.

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Task: `q002.

Write your work out on paper.

Don't try to do multi-step problems on your computer keyboard.

• It's quicker to write them out then transcribe your work on the keyboard, and the act of writing things down has a number of advantages.

• Writing things on paper allows you to organize your thoughts, to make multiple representations of the situation, and to save your work for reference.

• Writing, sketching, doodling, etc. also tend to reinforce the learning process.

Use sketches:

• Make sketches to represent the things you are thinking about and try to organize your thoughts as you proceed.

Take notes:

• You should always make notes as you work. Taking notes reinforces the learning process and provides you with a reference for the future.

• In some exercises a single complex problem or situation will be broken down into a series of questions. In such cases it will be necessary for you to maintain the thread of the problem. Maintaining at least brief notes will allow you to do so.

Please respond with a statement detailing your understanding of the advice given above.

**** Your response (insert your response beginning in the next line; the next line is blank and doesn't include the #$... prompt):

Organize thought on paper and then translate them onto keyboard into an understandable manner.

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Task: `q003.

On anything you send the instructor, including but not limited to q_a_ assignments, queries and tests, sufficient documentation is required to allow the instructor to follow your thinking and the details of your solution.

An example of good documentation to the question 'How long will it take to make $400 at $10 per hour?':

'At $10 / hour it will take $400 / ($10 / hour) = 40 hours to make $400.'

A poor answer to the same question:

'4000'

This is a poor answer first because it's undocumented, second because it's wrong, and third it can contribute to a habit of poor documentation, which will nearly always cost you points on your tests.

• It would be fairly easy for the instructor to figure out where the 4000 came from--most likely you multiplied when you should have divided, though you may have just been really careless with your 0's--so it might be possible to help you see what you did wrong here. However this is usually not the case with undocumented answers on more complicated problems.

• The more usual case is that your instructor has no clue about what you did wrong and no reasonable way to 'reverse-engineer' your solution and address your error.

• On a test the bad thing about such an answer is that even if you thought correctly through several steps and made only one minor error in your arithmetic, you didn't document the process and there would be no way to give you any partial credit.

Note also that if a question can be answered with 'true' or 'false' it doesn't matter whether you put down the right answer or not, if all you put down is 'true' or 'false' it is impossible to tell whether you got the answer by a correct process or by a coin flip, and in this course credit is not give for coin flips.

As another example, if a test problem asks for the graph of an expression it is not sufficient to copy the output of your graphing calculator; unless the problem specifically tells you to use the graphing calculator you must document how the characteristics of the graph result from the given expression. Document your answers, show the instructor that you know why the answer is what it is, or you risk getting no credit for the question.

Explain why it's important for you to document your work.

**** Your response (insert your response beginning in the next line; the next line is blank and doesn't include the #$... prompt

Explain to your instructor how you arrived at your answer documenting your work.

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Task: `q004.

To repeat something that will be especially important on tests:

• You cannot assert one of a limited number of answers and expect to receive credit (e.g., by choosing 'true' on a question to which the answer is 'true' or 'false').

• You must fully justify any answer, and especially answers for which a limited number of choices is possible.

• This means that you need to explain your thinking and show the steps of your solution.

Please explain what it means to justify an answer on a test, and why this is important:

**** Your response (insert your response beginning in the next line; the next line is blank and doesn't include the #$... prompt):

Explain and show your work.

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Task: `q004.

Don't waste your time misrepresenting what you know.

• Some students copy the homework of other students or receive inappropriate help on homework.

• Some students go through a q_a_ or query program and look at the answers, then essentially copy the answers on the next run.

The instructor notices this pattern but doesn't penalize it, and some students get the 10% or 15% of their grade that's based on homework and daily assignments in this manner. However students who use this strategy tend not to learn the material well and almost never succeed on the tests that make up the vast majority of their grade.

Please state these ideas in your own words.

**** Your response (insert your response beginning in the next line; the next line is blank and doesn't include the #$... prompt):

Your test will reflect exactly how you understand the assignments.

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Task: `q005.

Also it isn't strictly necessary to do all the homework and daily assignments, since test (and for physics students lab) grades are the dominant factors in your final grade. Some students do indeed succeed without submitting much work other than tests (and for physics courses labs).

However, while this is possible, it is strongly recommended that you DO NOT expect to be able to prepare for tests (and, where applicable, labs) without submitting the assignments.

• For most students it is simply necessary to go through the process and learn the material by submitting the assignments and getting instructor feedback.

Please state this in your own words.

**** Your response (insert your response beginning in the next line; the next line is blank and doesn't include the #$... prompt):

Following up with the assignments and instructor feedback will improve your score.

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Task: `q006.

When documenting test items you need to use the methods appropriate to your course.

• Just because you can get the right answer in one way or another does not mean that you are using a procedure on which you can build further understanding.

• It's not a matter of 'my way' vs. 'your way'. The structure of the subject dictates the things you need to understand.

• If you are taking a test on material which requires you to write and solve equations, for example, then using trial and error to arrive at even a correct solution is not valid and would not receive credit.

State this policy in your own words.

**** Your response (insert your response beginning in the next line; the next line is blank and doesn't include the #$... prompt):

Show your work and how you arrive at the answer.

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Task: `q007. It is also not valid to justify a solution by copying a picture or a solution from a calculator (unless of course the problem specifies that the calculator is to be used in this manner).

• It is fine to use a calculator to do your arithmetic, but you must, for example, show the steps in solving an equation.

• If the solution of a problem includes a graph you must explain the behavior of that graph rather than just copying calculator output.

The key is that while a calculator can be very useful, operations like entering a function or an equation and copying output from a calculator is not a college-level skill.

If the process is part of the course, you have to show the steps of the process.

State this policy in your own words.

**** Your response (insert your response beginning in the next line; the next line is blank and doesn't include the #$... prompt):

Show your work and how you arrived at the answer.

#$&* (your response should have gone on the line above this one)

Now highlight and copy your document, paste your copy into the box below.

Double-check to be sure you have accurately filled in the information at the beginning of this form.

Then click on Submit Form.

It is suggested that you save a copy of your document as a backup.

When you submit this document you will have complete Orientation Step 7. Your next activity will be Orientation Step 8.

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Self-critique (if necessary):

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course Mth 271

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.

001. typewriter notation

Note that there are six questions in this exercise. Be sure to continue scrolling down until you get to the end of the exercise.

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Question: `q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4). Then evaluate each expression for x = 2.

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Your solution:

You evaluate based on the expression. Therefor you would calculate what is in parenthesis first(follow order of operations).

x-2/x+4 = 5 and (x-2)/(x+4) = 0

confidence rating #$&*: 3

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Given Solution:

`aThe order of operations dictates that grouped expressions must be evaluated first, that exponentiation must be done before multiplication or division, which must be done before addition or subtraction.

It makes a big difference whether you subtract the 2 from the x or divide the -2 by 4 first. If there are no parentheses you have to divide before you subtract. Substituting 2 for x we get

2 - 2 / 2 + 4

= 2 - 1 + 4 (do multiplications and divisions before additions and subtractions)

= 5 (add and subtract in indicated order)

If there are parentheses you evaluate the grouped expressions first:

(x - 2) / (x + 4) = (2 - 2) / ( 2 + 4 ) = 0 / 6 = 0.

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Self-critique (if necessary): O.K

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Question: `q002. Explain the difference between 2 ^ x + 4 and 2 ^ (x + 4). Then evaluate each expression for x = 2.

Note that a ^ b means to raise a to the b power. This process is called exponentiation, and the ^ symbol is used on most calculators, and in most computer algebra systems, to represent exponentiation.

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Your solution:

Follow order of operations: 2^x+4 = 8 and 2^(x+4) = 64

confidence rating #$&*: 3

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Given Solution:

`a2 ^ x + 4 indicates that you are to raise 2 to the x power before adding the 4.

2 ^ (x + 4) indicates that you are to first evaluate x + 4, then raise 2 to this power.

If x = 2, then

2 ^ x + 4 = 2 ^ 2 + 4 = 2 * 2 + 4 = 4 + 4 = 8.

and

2 ^ (x + 4) = 2 ^ (2 + 4) = 2 ^ 6 = 2*2*2*2*2*2 = 64.

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Self-critique (if necessary): O.K

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Question: `q003. What is the numerator of the fraction in the expression x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x? What is the denominator? What do you get when you evaluate the expression for x = 2?

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Your solution:

Follow order of operations: Numerator = 3, the number is before the /(division sign)….order of operations. For the denominator follow the order of operations again. Denominator = 7 I utilized scratch paper to work problem and then type my results out. You can’t get in a hurry on these problems.

confidence rating #$&*: 3

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Given Solution:

`aThe numerator is 3. x isn't part of the fraction. / indicates division, which must always precede subtraction. Only the 3 is divided by [ (2x-5)^2 * 3x + 1 ] and only [ (2x-5)^2 * 3x + 1 ] divides 3.

If we mean (x - 3) / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x we have to write it that way.

The preceding comments show that the denominator is [ (2x-5)^2 * 3x + 1 ]

Evaluating the expression for x = 2:

- 3 / [ (2 * 2 - 5)^2 * 3(2) + 1 ] - 2 + 7*2 =

2 - 3 / [ (4 - 5)^2 * 6 + 1 ] - 2 + 14 = evaluate in parenthese; do multiplications outside parentheses

2 - 3 / [ (-1)^2 * 6 + 1 ] -2 + 14 = add inside parentheses

2 - 3 / [ 1 * 6 + 1 ] - 2 + 14 = exponentiate in bracketed term;

2 - 3 / 7 - 2 + 14 = evaluate in brackets

13 4/7 or 95/7 or about 13.57 add and subtract in order.

The details of the calculation 2 - 3 / 7 - 2 + 14:

Since multiplication precedes addition or subtraction the 3/7 must be done first, making 3/7 a fraction. Changing the order of the terms we have

2 - 2 + 14 - 3 / 7 = 14 - 3/7 = 98/7 - 3/7 = 95/7.

COMMON STUDENT QUESTION: ok, I dont understand why x isnt part of the fraction? And I dont understand why only the brackets are divided by 3..why not the rest of the equation?

INSTRUCTOR RESPONSE: Different situations give us different algebraic expressions; the situation dictates the form of the expression.

If the above expression was was written otherwise it would be a completely different expression and most likely give you a different result when you substitute.

If we intended the numerator to be x - 3 then the expression would be written (x - 3) / [(2x-5)^2 * 3x + 1 ] - 2 + 7x, with the x - 3 grouped.

If we intended the numerator to be the entire expression after the / the expression would be written x - 3 / [(2x-5)^2 * 3x + 1 - 2 + 7x ].

STUDENT COMMENT: I wasn't sure if the numerator would be 3 or -3. or is the subtraction sign just that a sign in this case?

INSTRUCTOR RESPONSE: In this case you would regard the - sign as an operation to be performed between the value of x and the value of the fraction, rather than as part of the numerator. That is, you would regard x - 3 / [ (2x-5)^2 * 3x + 1 ] as a subtraction of the fraction 3 / [ (2x-5)^2 * 3x + 1 ] from the term x.

STUDENT QUESTION: There was another question I had about this problem that wasn’t addressed. At the end when you changed the order of operation from

2 - 2 + 14 - 3/7 = 14 - 3/7

where did the 98/7 - 3/7 come into play before the end solution of 95/7? I must have forgotten how to do this part.

INSTRUCTOR RESPONSE: It's not clear how you can get 95/7 without this step.

To do the subtraction 14 - 3/7 both terms must be expressed in terms of a common denominator. The most convenient common denominator is 7.

So 14 must be expressed with denominator 7. This is accomplished by multiplying 14 by 7 / 7, obtaining 14 * 7 / 7 = 98 / 7. Since 7/7 = 1, we have just multiplied 14 by 1. We chose to use 7 / 7 in order to give us the desired denominator 7.

Thus our subtraction is

14 - 3/7 =

98/7 - 3/7 =

(98 - 3) / 7 =

95 /7.

STUDENT COMMENT

It took me a while to think thru this one especially when I got to working with the fraction. Fractions have always been my

weak spot. Any tips to make working with fractions a little easier is greatly appreciated.

INSTRUCTOR RESPONSE

Fractions are seriously undertaught in our schools, so your comment is not unusual.

I have to focus my attention on the subject matter of my courses, and while I do address it to a point, I don't have time to do justice to the subject of fractions. In any case , to do so would be redundant on my part, since there are a lot of excellent resources on the Internet.

I suggest you search the Web using something like 'review of fractions', and find something appropriate to your needs. You should definitely review the topic, as should 95% of all students entering your course.

STUDENT COMMENT

I think I am confused on why the Numerator is not the top portion and denominator the bottom portion of the problem.

INSTRUCTOR RESPONSE

Everything is on one line so there is no top or bottom in the given expression. A numerator and denominator are determined by a division of two expressions.

As we know, a denominator divides a numerator. In the given expression the division sign occurs between the 3 and the [ (2x-5)^2 * 3x + 1 ], so 3 is the numerator and [ (2x-5)^2 * 3x + 1 ] is the denominator.

x is not divided by the denominator, since the division occurs before the subtraction. For the same reason the -2 + 7x is not involved in the division. So neither the x nor the -2 + 7 x is part of the fractional expression.

STUDENT COMMENT

Didn’t know that 3 / 7 was 3/7 as a

fraction.

INSTRUCTOR RESPONSE

3/7 is treated as a fraction because of the order of operations. 3 must be divided by 7 before any other operation is applied to either number, and 3 divided by 7 is the fraction 3/7.

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Self-critique (if necessary): O.K

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Question: `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4.

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Your solution:

Follow order of operations:

Since^ comes first you evaluate that expression.

When I do the order of operations on paper I get (3+(3/4)-2) or 7/4

confidence rating #$&*: 3

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Given Solution:

`aWe get

(4-5)^2 * 4 - 1 + 3 / 4 - 2

= (-1)^2 * 4 - 1 + 3 / 4 - 2 evaluating the term in parentheses

= 1 * 4 - 1 + 3 / 4 - 2 exponentiating (2 is the exponent, which is applied to -1 rather than multiplying the 2 by 4

= 4 - 1 + 3/4 - 2 noting that 3/4 is a fraction and adding and subtracting in order we get

= 1 3/4 = 7 /4 (Note that we could group the expression as 4 - 1 - 2 + 3/4 = 1 + 3/4 = 1 3/4 = 7/4).

COMMON ERROR:

(4 - 5) ^ 2*4 - 1 + 3 / 4 - 2 =

-1 ^ 2*4 - 1 + 3 / 4-2 =

-1 ^ 8 -1 + 3 / 4 - 2.

INSTRUCTOR COMMENTS:

There are two errors here. In the second step you can't multiply 2 * 4 because you have (-1)^2, which must be done first. Exponentiation precedes multiplication.

Also it isn't quite correct to write -1^2*4 at the beginning of the second step. If you were supposed to multiply 2 * 4 the expression would be (-1)^(2 * 4).

Note also that the -1 needs to be grouped because the entire expression (-1) is taken to the power. -1^8 would be -1 because you would raise 1 to the power 8 before applying the - sign, which is effectively a multiplication by -1.

STUDENT QUESTION: if it's read (-1)^8 it would be 1 or would you apply the sign afterward even if it is grouped and it be a -1?

INSTRUCTOR RESPONSE: The 8th power won't occur in this problem, of course, but you ask a good question.

-1^8 would require raising 1 to the 8th power, then applying the negative sign, and the result would be -1.

(-1)^8 would be the 8th power of -1, which as you see would be 1.

STUDENT COMMENT: I think it would be easier to visualize what your trying to raise to the exponent if you actually put parenthesis around the 2, that part seems to get tricky on the computer.

INSTRUCTOR RESPONSE: The expression was intentionally written to be misleading and make the point that, to avoid ambiguity, order of operations apply strictly, no matter what the expression looks like.

Normally, for clarity, the parentheses would be included. They aren't necessary, but when helpful it's a good idea to include them. You can, of course, have too many parentheses in an expression, making it harder than necessary to sort out. In practice we try to strike a balance.

The original expression was

(x - 5) ^ 2x-1 + 3 / x-2

White spaces make no difference in how an expression is evaluated, but they can help show the structure; e.g.,

(x - 5)^2 * x - 1 + 3 / x -2

is a visual improvement over the original. The * between the 2 and the x is not strictly necessary, but is also helpful.

((((x - 5) ^ 2)) * x) - 1 + (3 / x) - 2

verges on having too many parentheses at the beginning; it does help clarify the 3 / x.

STUDENT COMMENT

Although I read through your explanation and do see the point you are making, that 2x is actually 2 * x, I still think that

(-1) should be raised to 2x rather than 2. Kaking the answer -11/4, not 7/4.

INSTRUCTOR RESPONSE

When the expression (x - 5) ^ 2x-1 + 3 / x-2 is copied and pasted into a computer algebra system it is translated as

This notation is universal and unambiguous. Any deviation from strict interpretation (which does occur among some authors and among manufacturers of some calculators) tends to result in ambiguity and confusion.

STUDENT COMMENT

While I do understand what you are trying to relate, I will continue to make these mistakes on more than one occasion and will not penalize myself for not rewriting years of mathematics because of a syntax issue in an online class.

INSTRUCTOR RESPONSE

I don't penalize errors in typed notation when the intent is clear (though I will sometimes point out these errors), and when you take your tests you'll be writing them out by hand and this won't be an issue.

However this is not a syntax issue in an online class. This is the order of operations, as it has been since algebra was developed hundreds of years ago, and it's completely consistent with the mathematics you appear to know (quite well).

As stated here, if you use the wrong syntax in any computer algebra system, your expression will not be interpreted correctly. For this reason alone you need to understand the notation.

For this and other valid reasons you need to understand how the order of operations are represented in 'linear' fashion (i.e., 'typewriter notation') and to correctly interpret expressions written in this notatation.

Any mathematics that has been learned correctly is completely consistent with the order of operations and with the notation used in this course. If the mathematics you've learned was inconsistent with the order of operations (and I don't believe this is so in your case, but it is with many students), then you would need to adjust your thinking. Fortunately this is very easy to do. Interpret expressions literally, assume nothing, and everything works out.

You will also find that the notation quickly becomes easy to read and use, and that it expands your comprehension of all mathematical notation.

STUDENT COMMENT

I used -1^(2*4). I didn't realize that was doing multiplication before exponents. All of this typewriter notations seems ambiguous to me but I think that had I seen the expression in standard notation I would probably have made the same mistake in this instance. If I were writing this expression I would probably use a parenthesis or * to show the necessary separation.

INSTRUCTOR RESPONSE

Parentheses, even when they aren't strictly necessary, are often useful to clarify the expression. An parentheses, even when not necessary, are part of the order of operations.

Spacing is not part of the order of operations. An expression has the same meaning even if all spaces are removed.

However as long as an expression is correctly formed, spacing as well as parentheses can certainly be used to make it more readable.

I don't go to any trouble in this exercise to make the expressions readable, since my goal here is to make the point about order of operations, which give an expression its unambiguous meaning.

However in most of the documents you will be working with, I do make an effort to clarify the meanings of expressions through their formatting, often using unnecessary parentheses and spacing to help clarify meaning.

Certainly I encourage you to do the same.

STUDENT QUESTION

I didn’t separate the Ύ as a stand alone fraction, I am confused about why you don’t treat it as an equation that the

denominator isn’t treated as a denominator.

INSTRUCTOR RESPONSE

Your work was good throughout most of this problem. You did forget to copy down a -1 in one of the early steps, but otherwise followed the order of operations correctly until nearly the last step.

However near the end you said that 4+3/4-2=7/2.

You appear to have performed the addition 4 + 3 and the subtraction 4 - 2 before dividing. However the division has to be done first.

The division sign is between the 3 and the 4, so the division is 3/4, and that gives you the fraction 3/4.

Therefore the expression 4+3/4-2 tells you to 'add 3/4 to 4 then subtract 2'.

When actually writing this out we would probably include parentheses. That wasn't done here, as it would have defeated the point being made about order of operations, but for clarity we might have written

4 + (3/4) - 2.

The parentheses are not necessary around the 3/4, since the order of operations is sufficient to unambiguously define the result, but they do make the expression easier to read and reduce the likelihood of error.

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Self-critique (if necessary): O.K

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Question: `q005. At the link

http://www.vhcc.edu/dsmith/genInfo/introductory problems/typewriter_notation_examples_with_links.htm

(copy this path into the Address box of your Internet browser; alternatively use the path

http://vhmthphy.vhcc.edu/ > General Information > Startup and Orientation (either scroll to bottom of page or click on Links to Supplemental Sites) > typewriter notation examples

and you will find a page containing a number of additional exercises and/or examples of typewriter notation.Locate this site, click on a few of the links, and describe what you see there.

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Your solution:

It gives you samples as to what expressions look like when you write them out on paper and brak them down into an understandable manner.

confidence rating #$&*: 3

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Given Solution:

`aYou should see a brief set of instructions and over 30 numbered examples. If you click on the word Picture you will see the standard-notation format of the expression. The link entitled Examples and Pictures, located in the initial instructions, shows all the examples and pictures without requiring you to click on the links. There is also a file which includes explanations.

The instructions include a note indicating that Liberal Arts Mathematics students don't need a deep understanding of the notation, Mth 173-4 and University Physics students need a very good understanding,

while students in other courses should understand the notation and should understand the more basic simplifications.

There is also a link to a page with pictures only, to provide the opportunity to translated standard notation into typewriter notation.

end program

STUDENT COMMENT (not quite correct)

I see a collection of typewriter problems, after looking at some of them I see that the slash mark is to create a fraction rather than to denote division.

INSTRUCTOR CORRECTION

A fraction is a division of the numerator by the denominator. The slash mark indicates division, which can often be denoted by a fraction.

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Self-critique (if necessary): O.K

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Question: `q006 Standard mathematics notation is easier to look at; it's easier to see the meaning of the expressions.

However it's very important to understand order of operations, and students do get used to this way of doing it.

You should of course write everything out in standard notation when you work it on paper.

It is likely that you will at some point use a computer algebra system, and when you do you will probably have to enter expressions using a keyboard, so it is well worth the trouble to get used to this notation.

Indicate your understanding of why it is important to understand this notation.

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Your solution:

I strongly agree that it is very very difficult to work order of operations out in your mind without writing them out on paper. It is best to write these out on paper and work the problems. Less chance fore errors.

confidence rating #$&*: 3

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Self-critique rating:

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