Writing Fractions “Correctly”

I very often see students writing, for example, $1/3x$ when they mean $\frac 13x$. I used to tell them not to write $1/3x$ beause it looks like $\frac{1}{3x}$ until I realized that, according to PEMDAS, it really does mean $\frac 13x$. I feel that using the vinculum (e.g. $\frac 13$) is easier on the mind than using the solidus, (e.g. $1/3$) but telling a student that seems to be the equivalent of "because I said so." Does anyone have a better reason for using one over the other?

• Interesting question. If you have a book with examples/answers using vincula (sp.?) then you could point to that and say we'd like to synchronize with the book expressions. – Daniel R. Collins Nov 3 '17 at 14:50
• @whatever: Interestingly, Mathjax answered that for you: make the outer one bigger. – Daniel R. Collins Nov 3 '17 at 18:13
• @DanFox It's a mnemonic for order of operations (parethneses, exponents, division, multiplication, addition, subtraction) – Peter Olson Nov 3 '17 at 20:29
• Ask them to evaluate 1/3x9. Hopefully half of the class will say "3" and the other half will say "1/27." Then you say "the correct answer is ..." Then hopefully half of the class will get upset and try to defend their answer. Then they will see that this way of writing is ambiguous and leads to multiple answers despite there being one correct answer. So now there is need for a way to write the expression without ambiguity (aka, the viniculum or parenthesis). – ruferd Nov 3 '17 at 20:50
• I think the problem here is that, despite what your students may have been taught, PEDMAS is not the emperor of the universe, whose word is unbreakable law. Lots of us (rebels?) parse formulas using only common sense and therefore find $1/3x$ ambiguous. – Andreas Blass Nov 3 '17 at 23:15

I think that, depending on the maturity level of the students, you could just talk to them about why writing $\frac{1}{3}x$ rather than $1/3x$ makes it much clearer what you mean. They should understand.

Remind them that the purpose of writing anything, including mathematics, is to clearly convey an idea to the reader, and the notation $1/3x$ is ambiguous: some people might read it as $(1/3)x$, but many others will read it as $1/(3x)$. The most straightforward way to avoid that lack of clarity is to just write $\frac{1}{3}x$ instead.

• I think some of the problem is that students get so wrapped up in applying the rules for transforming variable expressions that they lose sight of what the rules mean and what a variable expression represents. They over abstract the abstractions and get lost in a sea of rules and symbols. – Steven Gregory Nov 18 '17 at 16:43
• What would be the recommendation if restricted to unicode with no special formatting? (i.e. they're discussing it in an email or on fcebook?) Unicode has some fractions, such as ⅓, and you can use subscript and superscript to make pretty much any number, ²²/₇ but few are willing to go to that length, preferring shorthand. (2^2 as opposed to 2²) – DukeZhou May 15 '18 at 20:53
• @DukeZhou If you're writing things in plain-text and don't wanna bother with trying to format it (whether with Unicode characters, HTML markup, Markdown, etc) then you've just got to be careful to be clear. Because 1/3x is ambiguous, consider wrapping things in parenthesis to make it clear you mean (1/3)x. Of course if you have to write things linearly like this, you can start you get creative with your expressions. Why not write it as x/3? That's equivalent and perfectly unambiguous. – Mike Pierce May 16 '18 at 0:18

$1/3x = \frac{1}{3} x$ is the standard. The advantage of following the standard is that other people who also know the standard will understand what you write, and (most) calculators will do what you expect. Also, this is consistent with how addition and subtraction work, so there's less cognitive overload.

$1/3x = \frac{1}{3x}$ has the advantage that the typesetting better aligns with our spatial recognition abilities; i.e. the spacing delineates how the terms group. Also, in unformatted text, it can be a convenient shorthand.

What you should tell the students is "don't use this notation, no matter which way you mean it", since the goal of writing is to communicate your meaning, and you're doing a poor job of that if you choose to write in a way that is known to be commonly misunderstood.

• +1 for addressing the advantage of the 1/3x notation, which explains why it is such a common (albeit misleading) shorthand. – G Tony Jacobs Nov 13 '17 at 17:35
• Totally disagree that $1/3x=(1/3)x$. It's ambiguous. – Ben Crowell Dec 8 '17 at 4:52
• When something is ambiguous, look at it as if you were a calculator. – user7171 Dec 14 '17 at 13:56

Ask them the meaning of $x^2/3x$. What is its value when $x=2$?

The purpose of notation is to express an idea so that we or others can later understand it. By using notation in the example above, we have utterly defeated that purpose.

Either we meant $(x^2/3)x$ or we meant $x^2/(3x)$, so writing with a solidus requires brackets regardless. Note that the inclusion of a multiplication sign does not help, since that is implied by the convention of concatenation already, but it is unlikely that you or the students use the convention that concatenation implies multiplication and simultaneous brackets over the concatenated expression.

Using a vinculum makes it clear immediately: $\frac{x^2}{3}x$ versus $\frac{x^2}{3x}$.

This is an issue of what appears to be concatenation, but what we assume is multiplication. That is, placing $x$ next to the $3$ typically means "multiply", so we give it a place it in the regular order of operations. That being the case, the assumption is that what the author means by $1/3x$ is $1 \div 3 \times x$, which would be done from left-to-right. This is equivalent to the use of the vinculum: $\frac{1}{3}x$

However, there are definitely times where we concatenate but don't follow the order of operations. Consider the expression $$15 ft^2 \div 5 ft$$ where (I bet) the assumed meaning is $\frac{15 ft^2}{5 ft} = 3 ft$. But considering that the concatenation $5 ft$ could mean $5 \times ft$ or $ft \times 5$, then the original expression (obeying the order of operations) is ambiguous: $$15 ft^2 \div 5 ft = 15 \times ft \times ft \div 5 \times ft = 3 ft^3$$ Or... $$15 ft^2 \div 5 ft = 15 \times ft \times ft \div ft \times 5 = 75 ft$$

And, this is just the situation you are trying to prevent by cautioning your student to write $\frac{1}{5}x$ instead of $1/5x$. The notation they use should help clarify their meaning. [Incidentally, this issue isn't particular to units -- some calculators behave differently when concatenating a number with a parenthesis, and others when concatenating two variables (type "xy/x" on a TI-89).

The short answer I give my students is to treat concatenation with care. If you mean "multiply", then it is often best to use the symbol. We break this rule when simplifying algebra, writing $6x$, but always treating it as multiplication. I never model something like $1/6x$ without using a multiplication symbol, such as $1/6\cdot x$. So, if a student comes to me with $10/3x$, I will tell them that I honestly cannot tell what this means, and that they need to include a multiplication symbol, or parentheses or both.

• I think it has something to do with being taught to write "between the lines". Writing $\dfrac 13$ takes up a lot of vertical space and I think a lot of students have a conditioned aversion to doing that. – Steven Gregory Nov 4 '17 at 2:01
• Your example should reduce to 5t, not 5ft. – Nij Nov 12 '17 at 9:43

Division symbol ÷ Or / mean only 1 thing Left of operator is numerator. Right of operator is denominator.

Thus in 1/3x it is clear that denominator is product 3x.

However if one wants to write fraction(1/3)*x. They must simply put () around fraction as I have written or simply write x/3. This clears any ambiguity in the expression.

3x is a product (3)(x) no matter where it is in expression. So you were initially right to correct your students.

"1/2 of x" cannot be written as 1/2x But as x/2 As x must be in numerator and not in denominator. For the product x * 1/2 is evaluated correctly as x/2. Especially if brackets are not used.

"3/4 of x" as x * 3/4 or 3x/4 or use brackets (3/4)x Or (x)3/4.

All of these can be successfully evaluated using PEMDAS/ BODMAS without ambiguity.

One rule while writing fractions is to keep numerator in numerator and denominator in denominator and not mix them up.

• according to the rules of order of operation, $1/2x$ is to be interpreted as $(1/2)x$ – Steven Gregory Dec 7 '17 at 0:45
• This doesn't explain anything about the vinculum versus the solidus, it just describes how to make the solidus unambiguous, which is the entire problem in the question. – Nij Dec 7 '17 at 5:38
• Rules of order of operations give priority to assumed "fraction terms" over a Division operation ? When was this rule established in rules of order of operations? What rule of fraction supports writing x in denominator in an in-line expression for "Half x"? – K.Sheetal Dec 7 '17 at 11:17
• It will be much easier and correct if Fraction terms are strictly put in parenthesis when writing in-line expressions as rules of both division and fractions as well as order of operations support this. – K.Sheetal Dec 7 '17 at 11:37