Alg. I: Should answers be written as mixed numbers or improper fractions?

[Jen the RD]

My dd is using Lial's Intro Alg. So far, all fractional type answers in both the TM and solutions manual are being written as improper fractions. This is backwards from how I was taught. Simplification was to always include changing an improper fraction to a mixed number. Has that rule changed or is this simply the way Lial presents it? Which is correct?

 

Thanks,

Jennifer

[regentrude]

There is no "correct" or "incorrect" because each is the same number.

 

For all practical purposes, it is much easier to work with an improper fraction than with a mixed number which has to be converted before it can be used for anything - so if you want to USE the number in a further calculation (or if it occurs in a physics problem, for example), improper fraction is much more useful.

 

ETA: I can not recall any of our math texts algebra 1 and up to have converted to mixed numbers

[Jen the RD]

There is no "correct" or "incorrect" because each is the same number.

 

For all practical purposes, it is much easier to work with an improper fraction than with a mixed number which has to be converted before it can be used for anything - so if you want to USE the number in a further calculation (or if it occurs in a physics problem, for example), improper fraction is much more useful.

 

ETA: I can not recall any of our math texts algebra 1 and up to have converted to mixed numbers

 

This is very helpful. I certainly agree that it is much easier to use improper fractions rather than mixed numbers and I know dd feels that way! LOL! It's just funny that my teachers & profs in HS & college drilled into my head to always answer with a mixed number. They would count it wrong in improper fraction format. I wonder if standardized tests are nit-picky about this?

[kiana]

I have never seen a standardized test which included, for example, a) 1 and 3/4, b) 7/4 and expected you to choose the correct one. I'm not saying they don't exist, jut that I don't think you need to worry about that.

 

A mixed number CAN be useful when you want to know roughly how big something is, or if it's something like buying fabric by the yard. Usually I'd leave it as an improper fraction.

[Chris in VA]

I was taught to convert to mixed numbers, too, for a final answer--but then, I only went up to Algebra II. :D

[Dana]

I tell students to leave answers as improper fractions unless it's a word problem.

[Jen the RD]

OK, guys, thanks for the replies. I guess we'll follow what Lial's is doing and answer in improper fractions.

 

Jennifer

[WishboneDawn]

I've noticed that in Singapore DM answers are converted to mixed numbers so that's what we do.

 

I think it's a sneaky way for the text to get the kids to do one more bit of math for more practice. :)

[jennynd]

I guess it is the same number. However it does feel incomplete.. DS always convert because that is how i was taught.

[Candid]

The best answer is to go with whatever the checking authority is going to use. It's something to ask in classes. Besides the whole improper vs mixed number problem at some point you also have to decide if a decimal number is the better way to go as well.

 

I checked my favorite teaching math source, Professor Wu at Berkley, because I remembered he'd addressed this in one of his papers and found this quote:

 

Equivalent fractions naturally brings up the issue of whether students should always reduce each fraction to lowest terms. Implicit in this statement is the assumption that every fraction is equal to a unique fraction in lowest terms. (While this assumption is true and believable, it is nonetheless the case that its proof is quite nontrivial, depending as it does on the Euclidean algorithm.) What is more pertinent is the fact that, as a fraction,

12/9 is every bit as good as 4/3, and in general

nk/nl is every bit as good as n/l. An insistence on always having a

fraction in its lowest terms is thus a preference but not a mathematical necessity. Moreover, it is sometimes not immediately obvious whether a fraction is in lowest terms or not, e.g., 68/51. A more flexible attitude towards unreduced fractions would consequently make for a better mathematics education for school students.

(note the formatting broke up some from the PDF, blame that on me not him.)

 

That's in this paper: http://math.berkeley.edu/~wu/NMPfractions4.pdf

[Hunter's Moon]

I was always told to convert to a mixed number, but as I got to higher level math, it was usually easier to stick with an improper fraction as further calculations were easier in this way. Improper fractions can also be used in Chemistry (Stoichiometry). Mixed numbers would drive me bonkers in upper-level math and science :001_smile:

[FaithManor]

I was always told to convert to a mixed number, but as I got to higher level math, it was usually easier to stick with an improper fraction as further calculations were easier in this way. Improper fractions can also be used in Chemistry (Stoichiometry). Mixed numbers would drive me bonkers in upper-level math and science :001_smile:

 

 

Absolutely. When it is a word problem, it is generally best to convert to a mixed number because we have a mental picture of on object, feet of lumber, meters distance, yards of fabric, kilograms of flour, etc. In these cases, a mixed number is actually the simplest way to think about the answer or use the information.

 

But, when it comes to higher level maths, chemistry, physics, etc. one cannot plug a mixed number answer back into the equation or use it in a subsequent equation in which the answer from problem one is used in question two. In higher math, the mixed number would have to be converted back into an improper fraction in order to work the problem. This adds an unnecessary step.

 

So, at algebra 1, I start teaching the kids to use mixed numbers for story problems, and leave the answers as improper fractions for regular linear equations. This prepares them to not "add" a step when they encounter chemistry.

 

Faith

[DarlaS]

My dd is using Lial's Intro Alg. So far, all fractional type answers in both the TM and solutions manual are being written as improper fractions. This is backwards from how I was taught. Simplification was to always include changing an improper fraction to a mixed number. Has that rule changed or is this simply the way Lial presents it? Which is correct?

 

Thanks,

Jennifer

 

Mine has both (eight ed.).

[Jen the RD]

These are all helpful and thought provoking replies to my question. I particularly liked the quote from the guy at Berkeley. I agree with him that the educational system's emphasis on reducing fractions to their simplest form is an unnecessary focus. As many of you have pointed out, imporoper fractions are more practical to work with in higher math & science and I certainly remember all that from my chem/biochem classes in grad school. So, for our purposes, I'll stick with improper fractions, except in word problems where the mixed number may be more beneficial in applying and understanding a real life example.

 

Thanks so much,

Jennifer

[LizzyBee]

I was taught in high school and college to convert from improper fractions to mixed numbers unless the problem specified otherwise. Failure to convert was a wrong answer because it was incomplete.

[Jann in TX]

The (cough) 'standard' rule is that an answer to an equation is left as an improper fraction (improper does NOT mean incorrect--it is talking about its form). When answering a word problem an answer written as an improper fraction would not make general 'sense'-- I need 9/4 gallons of paint... so the 'rule' is when you are answering a word problem (definite final answer) convert to a mixed numeral.

 

Improper fractions must be reduced-- no exceptions. Reducing has nothing to do with mixed numbers!

 

I'm a Math major and I have a huge collection of Math texts... this is the general consensus... majority rules (or if you are in a class--DO WHAT YOUR INSTRUCTOR TELLS YOU!).

 

An improper fraction can easily be inserted into ANY other equation. In most Science applications, the answer to an equation is inserted into another formula... a mixed number must be converted to an improper fraction for multiplication and division.

 

--

I so wish I could go back in time and re-teach those poor 5th graders who had the misfortune to have me as their math teacher right out of college! I drilled it into their little heads that improper fractions were INCORRECT-- and were always marked wrong... Once I started teaching Algebra I found this subject to be my bane the first weeks of class-- I now spend time 'unteaching' this directive!-- I had to do it in class to day as a matter of fact!

[Jen the RD]

The (cough) 'standard' rule is that an answer to an equation is left as an improper fraction (improper does NOT mean incorrect--it is talking about its form). When answering a word problem an answer written as an improper fraction would not make general 'sense'-- I need 9/4 gallons of paint... so the 'rule' is when you are answering a word problem (definite final answer) convert to a mixed numeral.

 

Improper fractions must be reduced-- no exceptions. Reducing has nothing to do with mixed numbers!

 

I'm a Math major and I have a huge collection of Math texts... this is the general consensus... majority rules (or if you are in a class--DO WHAT YOUR INSTRUCTOR TELLS YOU!).

 

An improper fraction can easily be inserted into ANY other equation. In most Science applications, the answer to an equation is inserted into another formula... a mixed number must be converted to an improper fraction for multiplication and division.

 

--

I so wish I could go back in time and re-teach those poor 5th graders who had the misfortune to have me as their math teacher right out of college! I drilled it into their little heads that improper fractions were INCORRECT-- and were always marked wrong... Once I started teaching Algebra I found this subject to be my bane the first weeks of class-- I now spend time 'unteaching' this directive!-- I had to do it in class to day as a matter of fact!

 

Jann, I was hoping you would respond because your posts about Lial's are part of what directed me to that alg. curric. in the first place. I couldn't help by chuckle at the part I highlighted. I'll bet there are many other teachers out there who would agree with you. So, why is it that in the lower grades this is drilled so much into kids' heads only to have to be "untaught" later??? :confused:

Thanks again,

Jennifer