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Here are three "cancellations" seen during algebra simplification, two of which are invalid.

(1) $\frac{x + 6}{6} = \frac{x+6\hspace-1.2ex\diagup}{6\hspace-1.2ex\diagup} = x$

(2) $\frac{6x + 1}{6} = \frac{6\hspace-1.2ex\diagup x + 1}{6\hspace-1.2ex\diagup} = x + 1$

(3) $\frac{6x}{6} = \frac{6\hspace-1.2ex\diagup x}{6\hspace-1.2ex\diagup}$

What is the correct student-understandable feedback for the first two errors?


For (1), I have tried:

  • Adding 6 does not cancel with dividing by 6; those are not inverse operations.
  • Adding 6 would cancel with subtracting 6: $x + 6 - 6 = x + 6\hspace-1.8ex\diagup - 6\hspace-1.8ex\diagup$
  • Dividing by 6 would cancel with multiplying by 6: $\frac{6x}{6} = \frac{6\hspace-1.2ex\diagup x}{6\hspace-1.2ex\diagup}$

However, this feedback implies that perhaps (2) should be allowed, since the 6 is certainly multiplied in (2), so why not divide by 6 to undo it?


For both (1) and (2), I have also tried:

The numerator contains two terms, both of which are divided by 6, and the incorrect cancellation unfairly deprives the other term of being divided by 6.

However, this way of thinking implies that perhaps we are robbing the $x$ of its division by 6 in (3).


What is the best feedback for error (1) and error (2)?

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    $\begingroup$ Another factoring error I've seen is students cancelling $\frac{6}{6x}$ to $x$. $\endgroup$
    – TomKern
    Feb 18 at 3:39
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    $\begingroup$ Which level of education is this? How are they getting the feedback? $\endgroup$
    – Tommi
    Feb 18 at 6:29
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    $\begingroup$ I'm not sure it matters what level it is, but I have encountered the errors in developmental (pre-college-level) courses and also in all levels of calculus. I'm giving the feedback on exams or other written work, but would be happy to hear how other types of feedback could help. $\endgroup$
    – Chris Cunningham
    Feb 18 at 15:42
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    $\begingroup$ But it does matter. It's an oversight for a Calculus student, just as a plain arithmetic error would be. While for a student learning/practicing algebra it's an error in the main content of what is being learned. $\endgroup$ Feb 19 at 23:19
  • $\begingroup$ My experience in calculus classrooms at (United States) 2-year colleges and 4-year universities tells me that the majority of calculus students are also in the category "learning/practicing algebra," but that isn't even the point; the point is that even when someone makes an "oversight," they still should receive the correct feedback that explains precisely why their error is an error. $\endgroup$
    – Chris Cunningham
    Feb 21 at 17:06

8 Answers 8

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My thought process says the issue here is a misunderstanding of stylistic conventions leading to improper order of operations.

A fraction like $\frac{a+b}{c+d}$ has implied parentheses: $(a+b)\div(c+d)$ or $\frac{(a+b)}{(c+d)}$.

PEMDAS tells us we do parentheses first, then our multiplication/division step. If we're canceling a factor in the numerator and denominator, it has to be after we've taken care of the parentheses. The important issues in the implied parentheses are addition/subtraction operations.

If we ignore the implied parentheses, it's conceivable that PEMDAS tells us to multiply, then divide, then add: $\frac{6x+1}{6}$
multiply $\Rightarrow 6\cdot x = 6x$
divide $\Rightarrow 6x\div 6 = x$
add $\Rightarrow x + 1 = (x+1)$

A fraction like $\frac{6x}{6}$ still has the implied parentheses: $(6\cdot x)\div 6$. But the associative and commutative properties of multiplication mean it doesn't really matter when we're just using them to group factors instead of addends.

$\require{cancel}(6⋅x)÷6$ $=(x⋅6)÷6$ $=x⋅6÷6$ $=x⋅\cancelto{1}{(6÷6)}$ $=x$

We can still partially cancel sums using the distributive property, but it doesn't neatly cancel out the whole factor.

$\frac{6x+1}{6}$ $=\frac{1}{6}⋅(6x+1)$ $=\frac{\cancel{6}x}{\cancel{6}}+\frac{1}{6}$ $=x+\frac{1}{6}$

There can also be problems with division and subtraction, since order of operations is trickier there, but the concept still works if we're careful. The rule here is that division and subtraction are commutative with other divisors and subtrahends, so if we group all the factors/addends on the left and all the divisors/subtrahends on the right, we can still manipulate the problem to find identities that cancel out.

$\frac{abc}{abc}$ $=(abc)÷(abc)$ $=a⋅b⋅c÷a÷b÷c$ $=a⋅b⋅c÷c÷b÷a$ $=ab\cancelto{1}{(c÷c)}÷b÷a$ $=a\cancelto{1}{(b÷b)}÷a$ $=a÷a$ $=1$

Because a fraction like $\frac{abc}{abc}$ has already effectively grouped factors on the top and divisors on the bottom, we can immediately cancel them regardless of position, since we know they can always be re-arranged as above.

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My feedback might be more like this: When you are not 100% sure that a step works, try it with numbers. If x=2, does your expression keep the same value before and after your cancellation?

Also, you didn't say which level course. I see mistakes like this even among calculus students, so I have a worksheet I ask them to do in class, called Algebra Temptations. Step 1 is to mark each "identity" True or False. They are all false. Most students don't see this. We then discuss them.

If you are teaching a course like beginning algebra, I'd recommend thinking about how to teach or reteach this topic. I do say things like 1*x=x, so if we can get our fraction in a form where we have a 1 (or n/n) times something, then the 1 can go away, but otherwise "canceling" changes the value and doesn't work.

I do not expect most students to carefully read and think about my written feedback. So it seems better to ask them questions than to bother trying to explain perfectly why their work doesn't make sense.

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    $\begingroup$ Excellent answer! To address your last point, you could require corrections. If the student writes $\frac{x+6}{6} = x$, you could circle it and write "Use $x=6$ to show me why this cancellation is invalid." You could even circle a few correct equivalences and ask for examples which suggest that it is valid (if they write $\frac{6x}{6} = x$ you could ask them, in their revision, to demonstrate that this is valid for x=2, 3, 4). $\endgroup$ Feb 18 at 12:13
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    $\begingroup$ I see a problem with "try it with numbers": that is great if you know that the cancellation is wrong in order to give a counter-example. However, if you are trying to find out (like a student might), picking an unlucky set of numbers leads you astray. Consider $a^b$ "equals" $b^a$ with $4, 2$. $\endgroup$
    – TAR86
    Feb 18 at 12:51
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    $\begingroup$ "... and avoid 2 when you are trying numbers, since $2*2=2+2=2^2=\cdots$" $\endgroup$
    – Chris Cunningham
    Feb 18 at 15:44
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    $\begingroup$ Discouraging students from trying some test cases is throwing the baby out with the bath water. At this stage, it's more important that they begin to really appreciate that math is a logically consistent set of facts and equivalences can be falsified with a single counterexample from a simpler domain they already understand. You could just reiterate that one set of numbers working does not prove anything... so try a few to get a level of comfort that what you're doing /may/ be correct. $\endgroup$
    – Steve
    Feb 18 at 17:59
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    $\begingroup$ At least I can suggest 3 instead of 2. (I just want them to avoid 0 and 1. But I can imagine an example where a wrong step would get a right result for 2. $\endgroup$
    – Sue VanHattum
    Feb 18 at 20:07
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I tell my students that they can only cancel factors, and tell them to factor the numerator and denominator (to remind them what a factor is).

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    $\begingroup$ +1 for brevity and precision. I think this is what the OP needs in his specific use-case. $\endgroup$ Feb 18 at 18:19
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    $\begingroup$ It seems to me like students are more likely to know how to factor and recognize if something is factorable than to know what it means to be a factor! $\endgroup$
    – TomKern
    Feb 19 at 1:13
  • $\begingroup$ Sounds reasonable, and if the student responds and says they don't understand the usage, that's an opportunity to address it. I think this comment in the grading is the sharpest trigger for a diagnosis. $\endgroup$ Feb 19 at 5:43
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    $\begingroup$ This is how I approach it, too. I would add that in college, these mistakes are ingrained as prior knowledge, bad habits to be eliminated. The student needs to recognize this and practice checking their canceling. Both the teacher and the student need to understand it is a habit and it takes time to learn to inhibit it. The teacher telling and the student understanding what's wrong is not enough to fix the problem. Both have to have patience while the student's brain learns to suppress the bad habit. Practice strengthens both the right neural networks and the inhibitory ones, making it easier. $\endgroup$
    – Raciquel
    Feb 19 at 16:10
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    $\begingroup$ @JosephSible-ReinstateMonica 's point stops me from accepting this answer; I would like to "cancel" all inverse operations: Sums with differences, exponentiation with logarithms, cubes with cube roots, etc. However, I think this is the closest to the correct answer anyone has gotten. $\endgroup$
    – Chris Cunningham
    Feb 21 at 17:01
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What is the correct student-understandable feedback for the first two errors? [emphasis added]

I am not sure there exists a singular answer to this.

Ideally,

the educational trajectory should have students doing and saying things like this.

Grade 3 student: "$\frac{14}{3}=4\frac{2}{3}$... this is more than 4 wholes but less than 5 wholes. Fractions are numbers which mostly reside between consecutive whole numbers." [Student shows the $\frac{14}{3}=4\frac{2}{3}$ with area diagrams, set diagrams, number lines, manipulatives, etc.]

Grade 5 student: "$\frac{12}{9}=\frac{12\times5}{9\times5}=\frac{10}{45}=\frac{12\times30}{9\times30}=\frac{12\div3}{9\div3}=\frac{4}{3} ...$. Every fraction has an infinite number of equivalent fractions. You generate them by dividing or multiplying both the top and bottom by the same number. *You do not generate them by adding or subtracting the same amount from the top and bottom." [Student shows this by generating examples with varied representations of the idea.]

Grade 10 student: "$\frac{ab}{ac}=\frac{b}{c}$. Fractions can only be simplified by cancelling common factors of the entire numerator and of the entire denominator. This generalizes the idea of equivalent fractions."

Based on that grade 10-level understanding, one can then start discussing, comparing, and contrasting structures.

Teacher: "Can you see the form $\frac{ab}{ac}$ in these three rational expressions? If so, how? Show me if you can and let's start testing simplifications empirically."

I'd consider this to be a mathematically coherent experience.

But the reality ...

is more like this.

Students of many ages: "I hate fractions. I HATE them. HATE. They're nothin' but weird crap stains on a page that have been hurting my brain for years. But I've noticed that if I follow even weirder rules, then teachers get off my back and I can return to more enjoyable aspects of life. So just tell me what the weird rules are, tell me if my answer's right, and I'll forget all of this when you walk away, but that's OK as long as I'm not working on fractions any more. And... OMG NO I DON'T WANT TO DO FRACTIONS WITH LETTERS IN THEM HOLY COW YOU MADE THEM EVEN WORSE."

In these cases, there are obviously no easy answers. The only road to real understanding here is to meet students where they're at on that ideal trajectory then work up the ladder of abstraction one rung at a time. But that may require ignoring current coursework on which students are about to be graded. You want them to learn, but they want to get good grades NOW. And if they perceive rote cramming as the best way to get good grades, they'll do it, long-term consequences be damned.

Let me know if you are able to solve that one.

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  • $\begingroup$ The reality part of your answer just hit home with me! Reminds me of an email that I received from a student about 15 years ago (and it's still ingrained in my memory!). This student was in Calculus II (!!!, so she somehow had passed previous courses); and she emailed me after receiving a poor grade on the first midterm test, where in particular she complained that she "... found it very difficult to work on a test with fractions on it". $\endgroup$
    – zipirovich
    Feb 20 at 5:43
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I disagree with all your options because it is not solving the real issue, which is partly alluded to by BravoMath. This error only arises with students who were never correctly taught the true meaning of division and the meaning and purpose of precedence rules, so that is what you will have to do first, to solve the issue properly, and permanently. And of course the student must be taught the true meaning of multiplication before that, but I shall omit that from this post otherwise it would be too long.

~ ~ ~

Meaning of division

The meaning of "$x ÷ y$" is the quantity $z$ such that $x = y × z$. If there is no such $z$, then "$x ÷ y$" is meaningless. For example, "$6 ÷ 2$" means $3$ because $6 = 2 × 3$, and the meaning of "$1 ÷ 5$" is the quantity $s$ such that $1 = 5 × s$. For a concrete example of the latter, if you have $1$ pizza and divide it by $5$, the result is exactly the amount of pizza such that $5$ times that is $1$ pizza. Moreover, "$1 ÷ 0$" is meaningless, because there is no $z$ such that $1 = 0 × z$. At this level (elementary school), division means nothing else. (Do not even think about multiplicative inverses.)

Precedence rules

We need to communicate what we mean accurately to another person. This involves rules that we choose and agree to use. It is like natural language, where we both agree that if I say "A and B came" I am telling you that "A came" and also that "B came". Why? It's simply an agreement on a shared language, so that we know what each other is talking about. Mathematics is exactly the same; we agree on rules for what an arithmetic expression like "$1+2×3$" means. The rules have to tell us exactly what operations to do, on what, in what order. It is very important to explain all the non-mathematical things about agreement that I have said here!

(Note that you also need to teach what juxtaposition means and the rules for interpreting it. At this level, you can just say that you should insert multiplication symbols in-between juxtaposed numbers before reading it. But privately you should keep in mind that in actual mathematical writing it is not correct. Consider "$\cos 2x$" and "$\prod_{k=1}^n f(k) g(k)$" and "$\sum_{k=1}^n f(k) · \sum_{m=1}^n g(m)$". In case you are interested, one rule that seems to work for almost all conventions is that juxtaposition has higher precedence than everything else except brackets and postfix operations.)

~ ~ ~

In my teaching experience, not one student who understood the above two basic concepts ever made the mistakes you mentioned in your post (apart from careless errors). Why? Because the meaning of "$\frac{x+6}{6}$" is by definition the quantity $z$ such that $6 × z = x+6$. Certainly the student knows that "$6 × x$" is not the same as "$x+6$", so they cannot believe the wrong answer. Similarly, $\frac{6x+1}{6}$ is the quantity $w$ such that $6 × w = 6x+1$. Again, the student knows (from proper teaching of multiplication) that $6 × (x+1) = ( 6 × x ) + 6$, which is obviously different from $6 × x + 1$.

The point is, with the above definition of division, there is simply no cancellation. $\frac{6x}{6} = x$ because $6x = 6 × x$, and for no other reason.

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  • $\begingroup$ I like this answer a lot and I also like the answer about compression. The funny thing is they are the complete opposite of one another. This answer advocates that we should never "cancel": $6/6 = 1$ because $6 = 1\times 6$, and there is no other reason. I can't imagine actually computing anything this way, so I can't accept this answer, but I do think it is a good answer. $\endgroup$
    – Chris Cunningham
    Feb 19 at 19:05
  • $\begingroup$ @ChrisCunningham: You read it wrong. At the elementary level, the student must understand what division means by the definition. This doesn't mean that at the middle-school level, the student must not learn tricks to evaluate expressions correctly! Certainly, we accumulate a lot of tricks by the time we get to undergraduate-level, way beyond plain definitions, but that must never usurp the rightful place of definitions. $\endgroup$
    – user21820
    Feb 19 at 19:07
  • $\begingroup$ To put it in more striking terms, you can investigate various methods of computation but each one of them has to be judged for correctness against the definitions of the true meaning. Once students are taught to have that correct mindset, they will never go wrong, and will never be at any disadvantage in terms of computational facility either. In fact, they will be better off than other students because they can easily come up with their own computational tricks and check correctness all by themselves! $\endgroup$
    – user21820
    Feb 19 at 19:09
  • $\begingroup$ @ChrisCunningham: To make it explicit for the specific example in this thread, the cancellation happens in some cases and not others because of the meaning of the division. In particular, when you divide A by B where A = B × C, then A ÷ B = C by definition of division, so the cancellation of "B" in "( B × C ) / B" is entirely due to meaning, not computational procedure. Simplification of an expression should never be taught as something to be done, but something to be understood as equal. $\endgroup$
    – user21820
    Feb 19 at 19:22
  • $\begingroup$ "The meaning of "$𝑥\div 𝑦$" is the quantity $𝑧$ such that $𝑥=𝑦\times 𝑧$. If there is no such $𝑧$, then "$𝑥\div 𝑦$" is meaningless.... At this level (elementary school), division means nothing else." In more formal terms, all that means is that $\div$ is a partial function $(-)\div(-):(R \times R) \to (R + 1)$, where for any set of numbers $R$ and singleton $1$ and any pair of numbers $(x, y):R \times R$, $x \div y$ is either a number $z:R$ such that $y \times z = x$ or $\mathrm{meaningless}:1$. $\endgroup$
    – user19584
    Mar 1 at 18:08
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For these examples, you could also show some extended cancellation computations where they don't have to factor but just recognize that each fraction can be written as the sum of two pieces: $$ \begin{align} \frac{x+6}{6} &= \frac{x}{6} + \frac{6}{6} \\ &= \frac{x}{6} + 6 (\frac{1}{6}) \\ &= \frac{x}{6} + 1 \end{align}$$

or for the second example since they already understand the third example:

$$ \frac{6x+1}{6} = \frac{6x}{6} + \frac{1}{6} = x + \frac{1}{6}$$.

This way it's clear that "cancellation" is not some new operation with its own rules -- it is just a consequence of simpler rules we already know about fraction addition and multiplying numbers.

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  • $\begingroup$ Contrary to the "show a 1" suggestions, my understanding is that in the standard progression, basic canceling is actually more fundamental and comes first. It follows immediately from the definition of equivalent fractions. Personally I feel that being muddy about what was a definition and what was an implication held me back as an instructor. $\endgroup$ Feb 18 at 18:17
  • $\begingroup$ Breaking the fraction into two pieces can be misleading: $\frac{6x}{6} = 6 \frac{x}{6}$ implies no cancellation should occur here. But maybe that's okay. $\endgroup$
    – Chris Cunningham
    Feb 18 at 21:51
  • $\begingroup$ @ChrisCunningham Do your students view division by a number as equivalent to multiplication by its multiplicative inverse? $\endgroup$
    – Steve
    Feb 18 at 22:10
  • $\begingroup$ We are talking about students who can't decide whether or not to cancel various types of 6's; they will move the 6's around in a lot of ways. I don't know that they would phrase anything in that way. $\endgroup$
    – Chris Cunningham
    Feb 18 at 23:26
  • $\begingroup$ @ChrisCunningham ok in simpler terms as the same as multiplying by a fraction? $\endgroup$
    – Steve
    Feb 19 at 0:15
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Make them recall what $x$ means: substitute values (they do it, not you) and see that it's wrong. But this has already been said.

Another thing is use words. When students don't know how to read formulas, use plain, long, boring sentences: "take a number, add 6 to it and then divide the total by 6". They must be able to convert from sentences to formulas and viceversa. If they can't do that easily everything that follows will only be great frustration.

The motivation for this is: https://blogs.ams.org/matheducation/2019/09/30/is-there-a-switch-for-making-sense/

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    $\begingroup$ That's a great article; thanks for sharing it. $\endgroup$
    – ryang
    Feb 20 at 14:02
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    $\begingroup$ It goes so easily to a profound root of problems. I was shocked when I saw my students having real trouble discriminating the sentences that have a meaning from those that just don't make sense. Students lacking logic in words will have immense problems with logic in mathematics, readily apparent when complex formulas steal the scene from the bare calculations of elementary levels. $\endgroup$
    – user9137
    Feb 21 at 0:24
  • $\begingroup$ Yes cultivating the skill of crafting explanations (and descriptions) shouldn't be given such short shrift in mathematics classrooms; after all, precision in language is at the heart not just of applying mathematics but of mathematics itself, and this skill is neither particularly easy nor natural for most of us. More generally, developing the habit of making sense reaps dividends well outside mathematics and academia. $\endgroup$
    – ryang
    Feb 21 at 11:14
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What is the correct student-understandable feedback for the first two errors?

"Cancelling means dividing the numerator and the denominator by the same number. What do you get if you divide the whole term $6x+1$ by $6$?"

Either the student realizes the mistake or you are now at the true core of the problem: the student doesn't remember how sums are divided and this topic must be covered again.

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    $\begingroup$ Student: "$x+1$" $\endgroup$
    – Chris Cunningham
    Feb 19 at 17:00
  • $\begingroup$ OK, what's 6 +1 divided by 6? $\endgroup$ Feb 19 at 18:55
  • $\begingroup$ @MatthewTowers did you leave out the x on purpose? I still think that the proposed question in my answer is useful, I have added an explanation. $\endgroup$
    – Jasper
    Feb 20 at 11:08

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