BEDMAS incl. exponent: 9 x 6 / 2^3 / 4 = ...?

[steveopolis]

23/26 of these I got right, but three are throwing me, try as I might:

9 x 6 / 2³ / 4 =

What I'm doing;
2x2x2 = 8, so

9 x 6 / 8 / 4 =

9 x 6 =54
54 / 8 =6.75
6.75 / 4 = 1.6875

Answer key says 27/16. All the other answer key answers with fractions have fractions in the equation. I'm at a loss.

 

[ksdhart2]

You're probably going to feel real silly when you see it, but your answer and the book's answer are the same, just written differently. Consider your final answer in fractional form: \(\displaystyle \dfrac{6.75}{4}\). Now let's say you changed the denominator to be 16. What operation did you perform to do that? What if you did that same operation to the numerator too?

 

[Dr.Peterson]

23/26 of these I got right, but three are throwing me, try as I might:

9 x 6 / 2³ / 4 =

What I'm doing;
2x2x2 = 8, so

9 x 6 / 8 / 4 =

9 x 6 =54
54 / 8 =6.75
6.75 / 4 = 1.6875

Answer key says 27/16. All the other answer key answers with fractions have fractions in the equation. I'm at a loss.

You should develop the habit of working with fractions rather than decimals, as that is what math teachers tend to prefer (because fractions are always exact, whereas decimals often have to be rounded).

So 54/8 = 27/4 simplified; dividing this by 4 means multiplying by 1/4: 27/4 * 1/4 = 27/16.

If answers are not required to be fractions, though, you can just check that your answer is correct by converting their fraction to a decimal. Divide 27 by 16 and you get 1.6875. So, again, you were right.

 

[Steven G]

23/26 of these I got right, but three are throwing me, try as I might:

9 x 6 / 2³ / 4 =

What I'm doing;
2x2x2 = 8, so

9 x 6 / 8 / 4 =

9 x 6 =54
54 / 8 =6.75
6.75 / 4 = 1.6875

Answer key says 27/16. All the other answer key answers with fractions have fractions in the equation. I'm at a loss.

All the other answer key answers with fractions have fractions in the equation. I'm at a loss. But you do have fractions! 54 is being divided by 8, that is a fraction- 54/8. Then you are dividing that result, 6.75, by 4 getting another fraction.

After getting \(\displaystyle \frac{54}{8}\) you are dividing this by 4 which is the same as dividing by \(\displaystyle \frac{4}{1}\). But dividing by\(\displaystyle \frac{4}{1}\) is the same as multiplying by \(\displaystyle \frac{1}{4}\). Now \(\displaystyle \frac{54}{8}*\frac{1}{4} = 27/16\)

 

[steveopolis]

Interesting!!!

I'm still trying to understand how they arrived at it, though.

1. We're strictly taught that if the question is in decimals, we answer in decimals, and if the Q is in vertical fraction form, we answer in vertical fraction form. (And all answers in the book except this one follow that standard.)

2. We're taught to always put it in lowest terms, so I would never do the opposite it (and expect to get the mark).

3. I'm not sure how I would leap from "54 divided by 8 divided by 4" into suddenly deciding to make the first two factors into the format of a vertical fraction, and continuing the equation that way. What am I missing? Is this part of what's done, to suddenly (what seems randomly) decide to write a division equation in vertical fraction form and proceed from there?

I understand how it works mathematically, but I don't understand why we'd suddenly convert a linear division equation into a vertical fraction form and produce a vertical fraction form answer.

 

[j-astron]

I understand how it works mathematically, but I don't understand why we'd suddenly convert a linear division equation into a vertical fraction form and produce a vertical fraction form answer.

There is absolutely no difference between a division operation acting on two numbers, and the corresponding fraction. A fraction is just something divided by something else:

\(\displaystyle \displaystyle 54 / (8\times 4) = \frac{54}{8\times 4} \)

EDIT: you should take the above equality to mean that the two things on the left side and the right side mean exactly the same thing. They are just written slightly differently.

You end up with 54 over 32, and then you notice that both numerator and denominator are divisible by two (54/2 = 27 and 32/2 = 16), so you can reduce it into lowest terms

\(\displaystyle \frac{54}{32} = \frac{27}{16} \)

 

[Steven G]

Interesting!!!

I'm still trying to understand how they arrived at it, though.

1. We're strictly taught that if the question is in decimals, we answer in decimals, and if the Q is in vertical fraction form, we answer in vertical fraction form. (And all answers in the book except this one follow that standard.)

2. We're taught to always put it in lowest terms, so I would never do the opposite it (and expect to get the mark).

3. I'm not sure how I would leap from "54 divided by 8 divided by 4" into suddenly deciding to make the first two factors into the format of a vertical fraction, and continuing the equation that way. What am I missing? Is this part of what's done, to suddenly (what seems randomly) decide to write a division equation in vertical fraction form and proceed from there?

I understand how it works mathematically, but I don't understand why we'd suddenly convert a linear division equation into a vertical fraction form and produce a vertical fraction form answer.

What I have to say is similar to what j-astron had to say.

There is no difference between linear division and vertical fraction. In fact I never heard of linear division. What is true, that is equivalent, that is EXACTLY the same is that: x/y is exactly the same as \(\displaystyle \frac{x}{y}\)

 

[steveopolis]

I understand all of that (math), Jomo.

What I don't understand is when one might leap from "54 divided by 8 divided by 4" into suddenly deciding to make the first two factors into the format of a vertical fraction, and continuing the equation that way.

i.e., This equals that, yes, but there are different formatings within math. My teachers wrote an equation with no vertical form of fraction in it, then assumed we would morph formating part way through to supply a different format in the answer. (And they don't accept "either", despite being the same, unless they present both formats in the answer key.)

Or is it standard to leap to vertical format fractions whenever division comes up?

 

[steveopolis]

In fact I never heard of linear division.

I was referring to when we write it this way:

9 x 6 ÷ 2³ ÷ 4 =

...which is the way the equation was written, but I didn't know it would be important to present it this way for my post. My teachers have never asked me to reformat these into fractions before continuing the rest of the equation.

 

[j-astron]

Or is it standard to leap to vertical format fractions whenever division comes up?

Yes, it is, especially once you start doing algebra. Division is almost exclusively represented as fractions, and the inline division symbol (\(\displaystyle \div \) or / in plain text) is almost never used for hand calculations past elementary school, precisely because the inline division symbol can lead to ambiguity in terms of the order of operations (leading to the annoyance of having to memorize and apply arbitrary rules like 'BEDMAS'). Does 54/8*4 mean (54/8)*4 or 54/(8*4)? If you write it as a two-line fraction, there is never any ambiguity, because it's clear that

\(\displaystyle \displaystyle \frac{54}{8\times 4} \neq \frac{54}{8} \times 4 \)

As far as whether you're expected to make the "leap" to reformatting an answer as a fraction: in my opinion, at least, you're making a really big deal about distinctions and terminology/jargon (e.g. "vertical fraction format") that don't affect the underlying math. Distinctions that are a non-issue. If two ways of formatting an equation mean exactly the same thing, then, yes, they can be used interchangeably, and it is no big leap to do so. Try not to get too hung up on it.

As Dr. Peterson already suggested at the beginning of this thread, unless you're actually instructed to whip out a calculator, in general it's a good idea to leave final answers expressed in fractional form (in lowest terms) when doing arithmetic calculations by hand, because these answers are exact, whereas decimal representations can potentially be only approximate.

 

[Steven G]

23/26 of these I got right, but three are throwing me, try as I might:

9 x 6 / 2³ / 4 =

What I'm doing;
2x2x2 = 8, so

9 x 6 / 8 / 4 =

9 x 6 =54
54 / 8 =6.75
6.75 / 4 = 1.6875

Answer key says 27/16. All the other answer key answers with fractions have fractions in the equation. I'm at a loss.

Maybe this is what you are looking for. If a problem does not have decimals in it (initially), then you should not introduce decimals. If a problem has decimals, then I would leave the answer in decimal form. For example if someone asked me to add 2.12 and 7.31, I would leave the answer in decimal form. If some asked me to add $2.13 and $3.03, I would certainly not respond with $5 1/6 or $(31/6)

 

[steveopolis]

Yes, it is, especially once you start doing algebra. Division is almost exclusively represented as fractions, and the inline division symbol (\(\displaystyle \div \) or / in plain text) is almost never used for hand calculations past elementary school, precisely because the inline division symbol can lead to ambiguity in terms of the order of operations (leading to the annoyance of having to memorize and apply arbitrary rules like 'BEDMAS'). Does 54/8*4 mean (54/8)*4 or 54/(8*4)? If you write it as a two-line fraction, there is never any ambiguity, because it's clear that

\(\displaystyle \displaystyle \frac{54}{8\times 4} \neq \frac{54}{8} \times 4 \)

As far as whether you're expected to make the "leap" to reformatting an answer as a fraction: in my opinion, at least, you're making a really big deal about distinctions and terminology/jargon (e.g. "vertical fraction format") that don't affect the underlying math. Distinctions that are a non-issue. If two ways of formatting an equation mean exactly the same thing, then, yes, they can be used interchangeably, and it is no big leap to do so. Try not to get too hung up on it.

As Dr. Peterson already suggested at the beginning of this thread, unless you're actually instructed to whip out a calculator, in general it's a good idea to leave final answers expressed in fractional form (in lowest terms) when doing arithmetic calculations by hand, because these answers are exact, whereas decimal representations can potentially be only approximate.

Thank you, j-astron. Very helpful!!

Yes, I'm making a big deal (not out of jargon but) out of format, because the format of our answer adds or removes points in the assignments and tests. In other words, I wouldn't make a big deal out of a difference in format -but my teachers do! They mark according to their preference in method* and format, not only for accuracy in answer. The course is all over the map, though, so I'm often looking for consistency and sense where neither have yet been presented. (Ironically, I moved to math in a desire for precision, standard, etc, and have not found that in the course I'm doing.)

In this course, I'm learning two things: math, and what the teachers demand to see on paper at each stage. It helps me to understand what is standard and what is a quirk of the course.

In the last six months I've moved from being introduced to numbers, to basic addition of the smallest ones, through basic multiplication and division (i.e., the equivalent of elementary school), and onward up to this stuff. So, I'm only now being introduced to algebra, and we've been allowed to use a calculator only as of this newest segment.

Your information -(and Dr.Peterson's) that yes, as we move into algebra it is standard to go with fractions- is very helpful to me. I still don't know what the teachers will necessarily accept where, but this is good info. I'll watch for that.


*This came up in another topic, too. Jomo really really really does not want me to cross-multiply, but my teachers require that I show exactly that on the page. Many other basic things I'm able to do in my head, but I'm required to lay out a specific format to pass, even if there is a faster or smarter way to achieve the answer.

 

[Dr.Peterson]

Yes, I'm making a big deal (not out of jargon but) out of format, because the format of our answer adds or removes points in the assignments and tests. In other words, I wouldn't make a big deal out of a difference in format -but my teachers do! They mark according to their preference in method* and format, not only for accuracy in answer. The course is all over the map, though, so I'm often looking for consistency and sense where neither have yet been presented. (Ironically, I moved to math in a desire for precision, standard, etc, and have not found that in the course I'm doing.)

In this course, I'm learning two things: math, and what the teachers demand to see on paper at each stage. It helps me to understand what is standard and what is a quirk of the course.

In the last six months I've moved from being introduced to numbers, to basic addition of the smallest ones, through basic multiplication and division (i.e., the equivalent of elementary school), and onward up to this stuff. So, I'm only now being introduced to algebra, and we've been allowed to use a calculator only as of this newest segment.

Your information -(and Dr.Peterson's) that yes, as we move into algebra it is standard to go with fractions- is very helpful to me. I still don't know what the teachers will necessarily accept where, but this is good info. I'll watch for that.

It might be helpful if you tell us a little more about the format of the course. I have sometimes imagined it to be a computer-based course with no teachers you can ask; but that can't be true if your work, as well as your answer, is graded. If there is a teacher you can talk to, doing so could both clarify expectations and get scores adjusted. I forget what you've actually stated about the course; is it a remedial course at a college, or what? Who is doing the grading? Is there a textbook or other resource available, and how closely do the teachers follow it?

A good computer-based system will necessarily be picky about some things (because they have to program what answers are accepted), but should tell you precisely what is expected -- if they want a fractional answer, they will tell you that (though students often have to be reminded to read that part). A human teacher should likewise tell you the rules of the game, or else allow different answers when they do not go against the stated rules. If you were just going through a book and judging your own answer by what they say in the back, then you would need to be aware that the answer they give is not the only acceptable form.

In any case, consistency is important, and if each problem does not specify the required format, but grading demands one, then I would absolutely expect a general statement at the top of an assignment or a chapter telling you what is demanded. And you should be able to complain (respectfully ...) about unfairness.

 

[steveopolis]

It's an adult upgrading course -intended for those of us who dropped out of high school 30 years ago (often as a result of learning disabilities, like in my case), or just moved to the country (e.g., excellent math skill, but having to learn this school's approaches or the English terms for it all), etc. So it started out literally introducing numbers -very Sesame Street, which I very much appreciated

No computers, all paper. We learn from the books, on our own, at our own pace. So, no class instruction. Originally there were teachers to ask -and each of three teachers had their own approaches, methods, and preferences, so one would require that we disregard what the book says and do it differently (oh!), another would mark things wrong if not done to their personal preference, etc. Lots of ambiguity, confusion, frustration. So, I tried to learn *all* the ways, cover all my bases. That's worked, and I've achieved 98-100% on all tests and exams so far. When I lose that 1/2 mark or two marks to genuine error or to unidentified teacher preference, it doesn't take me down because I got the rest.

If I were to go simply with what the course presents, without pressing for better info, I'd be getting far, far lower marks.

The courses (text of explanations, exercises, answer keys, tests) are written by local faculty.

When I started, and noticed errors anywhere, I thought they would want to repair those. They didn't. They shrugged, saying, "Meh, sometimes things are wrong." I found this very frustrating, because having almost zero math knowledge, I have no way of knowing when an answer key is wrong or a question is missing some necessary info. Because I'm learning, I would assume I'm wrong and spend up to an hour working through a question over and over, reading back through the course to find what I wasn't getting, etc. Eventually a new teacher came on board and told me, "Don't do that -the course is often wrong." Oh.

We hand in paper assignments, but don't receive them back, so we don't know what we did right or wrong.

When the college semester ended, the teacher availability disappeared. Now it's a "welfare line" system: you wait in line up to 40 minutes, ask one question, go back to the end of the line, wait up to 40 minutes again to ask another question. This development within some other highly stressful circumstances had me close to quitting last week...then I found you guys!!! Seriously so grateful and relieved! I'd really wanted to learn math, and with your guys' help, I believe again that I can!

I know I asked a number of questions here this weekend, but I worked on the course for probably 12 hours Friday eve through Sunday eve, and completed dozens upon dozens on my own. Today I have two tests and woke feeling happy, confident, and peaceful because of your guys' help to me

 

[j-astron]

Thank you, j-astron. Very helpful!!

Yes, I'm making a big deal (not out of jargon but) out of format, because the format of our answer adds or removes points in the assignments and tests. In other words, I wouldn't make a big deal out of a difference in format -but my teachers do! They mark according to their preference in method* and format, not only for accuracy in answer. The course is all over the map, though, so I'm often looking for consistency and sense where neither have yet been presented. (Ironically, I moved to math in a desire for precision, standard, etc, and have not found that in the course I'm doing.)

In this course, I'm learning two things: math, and what the teachers demand to see on paper at each stage. It helps me to understand what is standard and what is a quirk of the course.

In the last six months I've moved from being introduced to numbers, to basic addition of the smallest ones, through basic multiplication and division (i.e., the equivalent of elementary school), and onward up to this stuff. So, I'm only now being introduced to algebra, and we've been allowed to use a calculator only as of this newest segment.

Your information -(and Dr.Peterson's) that yes, as we move into algebra it is standard to go with fractions- is very helpful to me. I still don't know what the teachers will necessarily accept where, but this is good info. I'll watch for that.


*This came up in another topic, too. Jomo really really really does not want me to cross-multiply, but my teachers require that I show exactly that on the page. Many other basic things I'm able to do in my head, but I'm required to lay out a specific format to pass, even if there is a faster or smarter way to achieve the answer.

That makes a lot of sense. Sorry you're stuck with arbitrary requirements, and apologies if I came off as harsh at all. You seem to have a good distinction in your head at this point between the fundamental principles and the rest of the course-based nonsense.

 

[j-astron]

When I started, and noticed errors anywhere, I thought they would want to repair those. They didn't. They shrugged, saying, "Meh, sometimes things are wrong." I found this very frustrating, because having almost zero math knowledge, I have no way of knowing when an answer key is wrong or a question is missing some necessary info. Because I'm learning, I would assume I'm wrong and spend up to an hour working through a question over and over, reading back through the course to find what I wasn't getting, etc. Eventually a new teacher came on board and told me, "Don't do that -the course is often wrong." Oh.

We hand in paper assignments, but don't receive them back, so we don't know what we did right or wrong.

When the college semester ended, the teacher availability disappeared. Now it's a "welfare line" system: you wait in line up to 40 minutes, ask one question, go back to the end of the line, wait up to 40 minutes again to ask another question. This development within some other highly stressful circumstances had me close to quitting last week...then I found you guys!!! Seriously so grateful and relieved! I'd really wanted to learn math, and with your guys' help, I believe again that I can!

I'm sorry. That seems like a really horrendous way to run a course. I don't know what self-respecting educator or mathematician would be okay with keeping incorrect or misleading solutions. The level of apathy here is demoralizing.

 

[steveopolis]

I'm sorry. That seems like a really horrendous way to run a course. I don't know what self-respecting educator or mathematician would be okay with keeping incorrect or misleading solutions. The level of apathy here is demoralizing.

It should be of comfort to us that none of them are mathematicians Their fields are very, very different from math (though they seem quite good at it regardless!).

Yes, "the level of apathy here is demoralizing." That's been my experience exactly! I've been in tears at points that they just didn't care about accuracy or standards or effective communications. The new one was, but he was quickly worn down by the Powers' disinterest. i.e., He remains keen and engaged, but he was told point blank that the errors would not be corrected, and that students would continue to be left to their best guesses.

I'm hopeful that once I reach college-level courses, I'll be using texts that have been checked, rechecked, vetted... and taught by people who are as passionate about accuracy as I am. Just gotta get there!!