Writing Fractions "Correctly"

Question

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?

Answer 1

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.

Answer 2

$1/3x = \frac{1}{3} x$ 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.

Answer 3

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 $\frac{x^2}{3x}$.

Answer 4

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.

Answer 5

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.