What is the right feedback for incorrect cancellation?

Question

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?

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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?

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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).

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

Answer 1

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.

Answer 2

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.

Answer 3

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).

Answer 4

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.

Answer 5

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.

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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.)

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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.

Answer 6

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.

Answer 7

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/

Answer 8

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.