Keeping, Changing, Flipping

Consider the following task: \[1.\quad \text{Simplify the expression }\frac{3}{4}\div\frac{2}{5}.\]

It is very common for students to struggle with this sort of task. A common teaching approach is “Keep Change Flip,” but too often that’s presented as a mechanical trick without any deeper understanding of why it works. In proper mathematical language, “Keep Change Flip” translates to “Dividing by a fraction is equivalent to multiplying by the reciprocal.”

One of the biggest confusions with this sort of exercise is that students forget which fraction to “flip,” which means they really don’t understand the process. I’d argue that it also shows they generally don’t know what fractions are.

---

Now consider the following task: \[2.\quad \text{Evaluate }(3 – 4) – (2 – 5).\]

The only common mistake that I’d anticipate here is that students would forget to distribute the middle subtraction. That is, I’d expect most students to evaluate one of these: \[3.\quad 3 – 4 – 2 + 5\\4. \quad 3 – 4 – 2 – 5\]

The first one is correct, the second one is incorrect. But it would greatly surprise me to see many students try to do something incorrect with 3–4.

From here, those students who know how to properly apply the order of operations would then work from left to right, yielding 2 (from (3)) or -8 (from (4)). Students who think addition is done before subtraction (thanks to a misapplication of PEMDAS) will get -8 from (3).

I’m going to toss out standard notation for a moment now. Let’s imagine that mathematical notation had evolved differently: There is no subtraction, and negative numbers are indicated with underbars. That is, (2) would be written as: \[5.\quad \text{Evaluate } (3 + \underline{4}) + \underline{(2 + \underline{5})}.\]

Underbars can be removed in pairs. So let’s evaluate (5) by first removing the parentheses, then removing unneeded underbars: \[6.\quad (3+\underline{4}) + \underline{(2+\underline{5})} = 3 + \underline{4} + \underline{2} + \underline{\underline{5}} = 3 + \underline{4} + \underline{2} + 5\]

Let’s put the numbers without underbars before those with, and then regroup with parentheses: \[7.\quad 3 + \underline{4} + \underline{2} + 5 = 3 + 5 + \underline{4} + \underline{2} = (3 + 5) + \underline{(4 + 2)} = 8 + \underline{6} = 2\]

Doing these steps using standard notation yields: \[8.\quad \begin{align}(3 – 4)-(2-5)&=3-4-2+5=3+5-4-2\\&=(3+5)-(4+2)=8-6=2\end{align}\]

This item is about division of fractions, though. How is this related to that?

---

Again, imagine mathematical notation has evolved differently: There is no division; divisors are indicated with overbars. There are no fractions, either. A fraction is indicated with multiplication by the divisor. That is, \[9.\quad \frac{3}{4} = 3 \times \overline{4}\]

So how would (1) be evaluated? Repeat the same steps as we did with the addition. Remove the parentheses and unneeded overbars: \[10.\quad (3 \times \overline{4}) \times \overline{(2\times\overline{5})} = 3 \times \overline{4} \times \overline{2} \times \overline{\overline{5}} = 3 \times \overline{4} \times \overline{2} \times 5\]

Put the numbers without overbars before those with, and then regroup with parentheses: \[11.\quad 3\times\overline{4}\times\overline{2}\times 5 = 3\times 5\times\overline{4}\times\overline{2} = (3 \times 5) \times \overline{(4\times 2)} = 15 \times \overline{8} = 1.875\]

Notice that, in our reimagined notation, addition/subtraction (6/7) works exactly like multiplication/division (10/11), as far as rewriting expressions is concerned. Likewise, our cancellation rules are parallel: \[12.\quad a + \underline{a} = 0 \\ 13.\quad a \times \overline{a} = 1\]

But that’s not our current notation, so I’ll rewrite (10) and (11) using that: \[14.\quad \begin{align} (3\div 4) \div(2\div 5) &= 3\div 4 \div 5 \times 5 = 3 \times 5 \div 4 \div 2 \\ &= (3 \times 5)\div (4\times 5) = 15 \div 8 = 1.875\end{align}\]

Compare (14) to (8). It is common to write things like (8), and to discuss that “subtracting a negative yields the same result as adding a positive,” but I don’t believe I’ve ever seen the same done with division as in (14).

These mechanisms are hidden by our standard fraction notation, which encourages us to just shortcut to “division by a fraction is multiplication by its reciprocal” or “Keep Change Flip.”

---

I introduced my alternative notation not as a suggestion for a serious replacement (our current symbols are far too entrenched for that) but to reinforce a perception of mathematical expressions involving operations are a string of pearls. We can rearrange our pearls at will. For instance, we could also write: \[15.\quad 3 \times \overline{4} \times \overline{2} \times 5 = 3 \times \overline{4} \times 5 \times \overline{2}\]

That is, \[16.\quad 3\div 4\div 2 \times 5 = 3\div 4 \times 5 \div 2 = (3 \div 4)\times (5\div 2)\]

Which is to say, \[17.\quad \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2}\]

More generally speaking: \[18.\quad \frac{a}{b}\div\frac{c}{d} = a\times \overline{b} \times \overline{c} \times d = a \times \overline{b} \times d \times \overline{c} = \frac{a}{b}\times\frac{d}{c}\]

---

This leads me to two controversial mathematical statements:

19. Multiplication is not just repeated addition

20. Fraction notation is just another way of writing division

When we say that multiplication is repeated addition, we get in the way of a very important concept: In standard arithmetic, over and over, we see that addition and multiplication behave in parallel ways, including with regards to their respective inverse operations. This paves the way to a broader discussion of inverse operations and parallel behavior: For instance, how does this behavior integrate with exponential and logarithmic functions? With differentiation and integration?

I saw a discussion on social media recently about the Quotient Rule in differentiation. Several people say they don’t even bother with it: They use the Power Rule and the Chain Rule, rewriting the divisor function as its multiplicative inverse by applying the exponent -1. Without a clear understanding of the relationship between division and subtraction, the understanding that \(\frac{1}{x} = x^{-1}\) is superficial, mechanical, and hence difficult to apply in a meaningful way.

And while there are practical applications in which a fraction is a meaningful concept in its own right, from a mathematical perspective (at least at and above the secondary level), fraction notation is most commonly used as a way of indicating division. I personally wonder the extent to which fractions confuse students because of the insistence at certain levels of insisting that they’re different from division.

---

In summary, then, I wonder the extent to which student problems with (1) are rooted in notation and the way in which fractions are presented. If the division isn’t transparent (that is, if students don’t see that there are three requests to divide here, not just one), then techniques for simplification will be mechanical applications of an algorithm instead of showing more understanding.

It is certainly more work to explain the deeper process, and to break student misconceptions about fractions being inherently different than multiplication, but how does that weigh against the time consumed in repeating, over and over, algorithms like “Keep Change Flip,” and how does it weigh against that in terms of depth of understanding?