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I am tutoring a high school level student (who is currently at the level of being introduced to anti derivatives) who has quite some trouble with grasping mathematics. We have been making good progress but then with this concept I get to a hault: he understands that $$\frac 22, \quad \frac 33, \quad \frac 44,\quad \ldots$$ all clearly are equal to one but as soon as I replace the number with a letter, $$\frac xx$$ he does not get it and starts guessing:

nothing? x? zero? one?

What is a good approach to teaching this?

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    $\begingroup$ It sounds like this student, despite being in a Calc or Pre-Calc class, has some major gaps in knowledge of earlier material. Have you tried to see what other, earlier concepts he struggles with? For example, can he solve for $x$ if given $x + 1 = -5$? Can he find all possible values for $x$ given $x^2 + 8 = 6x$? Also, looking at your list: Can he evaluate $\frac{1}{1}$? What about $\frac{-2}{-2}$? Can he explain what the expression here (note that you should specify $x \neq 0$) means? How is his understanding of division? Etc. $\endgroup$ – Benjamin Dickman Nov 18 '14 at 9:13
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    $\begingroup$ @BenjaminDickman, your comment should probably be an answer. I would make similar suggestions. $\endgroup$ – Sue VanHattum Nov 18 '14 at 14:45
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    $\begingroup$ i would be very concerned if i had a student in calculus class with derivatives/anti-derivatives and they did not truly understand x/x...great suggestion by @BenjaminDickman to really check for gaps in knowledge, it sounds like there are a lot of them and there is no sense with pushing through calculus with these gaps there $\endgroup$ – celeriko Nov 18 '14 at 17:50
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    $\begingroup$ Not an answer ... but I think that's what's refered to as concrete thinking .. if you would like to read up further. This person can't think abstract yet - maybe start looking into improving this person's abstract thinking - there is lots of techniques for this. $\endgroup$ – z0mbi3 Nov 20 '14 at 18:52
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I find that I can ask the following series of questions, which bridge the gap between "3" and "x."

What is $\frac22$?

What is $\frac33$?

What is $\frac66$?

What is $\frac{17}{17}$?

This is key -- 17 is a number that no one has any intuition about. If the student can deal with 17, they are already abstracting the idea as something that works with any number. If they get stuck before 17, then you know they are not successfully abstracting the idea at all, and you should go back to 5.

What is $\frac{179}{179}$?

Continue like this -- be ridiculous if needed, use 1578302, until you sense that the student is slightly annoyed by your questions because they definitely get the pattern. Then I like to go with

What is $\frac{\textrm{smiley face}}{\textrm{smiley face}}$?

If $x$ is a concept they are struggling with, then the $x$ itself might be threatening and might cause a mental block. Don't use $x$ itself. Use some other placeholder symbol. A picture of a bird from angry birds, if you want. They will probably be able to work with smiley face and get the answer because that is the same idea they have to use to work with 17. No one is actually imagining a pile of 17 things, divided into 17 piles. If they can do $\frac{17}{17}$, then they already get it, you just have to show them they already get it.

Then ask the big question:

Does it matter what I write here? [pointing at the top and bottom]

You basically already made it to the punchline which is that $\frac{x}{x}$ is 1. But saying that it is the same fact as $\frac{y}{y} = 1$ and $\frac{z}{z} = 1$ is a big help.

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    $\begingroup$ +1 for the smiley face reference. I had a professor use "cow" to get his point across. Smiley brings the approach into the 21st century. $\endgroup$ – JTP - Apologise to Monica Nov 18 '14 at 16:52
  • $\begingroup$ Interesting perspective; I had never considered the smiley-face to be a modern symbol! $\endgroup$ – Chris Cunningham Nov 18 '14 at 17:41
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    $\begingroup$ There are some useful extra problems you could give here like $\frac{2.79}{2.79}$ and $\frac{213+324}{213+324}$. $\endgroup$ – Steven Gubkin Nov 18 '14 at 19:32
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    $\begingroup$ This is indeed a great answer. By following the procedure suggested in this answer it didn't take much time before he conceptually understood the "$x/x$" problem. $\endgroup$ – Therkel Nov 23 '14 at 12:20
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    $\begingroup$ Itt did, however not initially solve his lacking intuition about fractions in general which showed when I asked: "what then is $\frac{2\cdot 179}{179}$?". We eventually worked this out by following the thought process from the second last paragraph the answer by @JPBurke. This led him to have an almost epiphany in the sense that finally he understood why something like the above fraction has an immediate and easy solution. $\endgroup$ – Therkel Nov 23 '14 at 12:39
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The problem may very well be that the student was expected to learn fractions and instead learned to manipulate symbols, but did not learn the meaning of a fraction. To reference part of Chris Cunningham's example, it certainly is possible to convey to a student that if the symbol on the top matches the symbol on the bottom ( i.e. something like $\frac{\textrm{smiley face}}{\textrm{smiley face}}$ ) that this is a kind of special case (which can evaluate to $1$). But an understanding grounded in meaning rather than just symbols is more likely to connect with the students greater body of mathematical understanding.

If the student doesn't see $\frac{3}{3}$ as "three parts of a whole that is composed of three parts" then this is an opportunity to address this understanding. (You could ask the student to describe what the $3$ in the denominator means, and the $3$ in the numerator, and see if they even grasp the underlying meaning of a fraction).

Once you've addressed the part-whole relationship meaning of fraction, which (even if neglected earlier in his education) should be comparatively easy to explain, since it is grounded in meaning, you can address the generalization of $\frac{x}{x}$.

"Assuming this represents a fraction, what does the $x$ in the denominator represent?" "Now, how many of $x$ pieces do we have in this so-called generalized fraction? How much of a whole does that make?"

The obvious problem here is that you're asking for a numerical answer, using fractions as examples, but $\frac{x}{x}$ can represent non-fractions like $\frac{0}{0}$, which you have not ruled out. So, depending on his understanding, your student may well be right to hesitate and, perhaps, wonder what you're getting at.

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    $\begingroup$ I'd like to second the remark about 😄/😄; I think this is not a great example, since, with an $x$, one is (ostensibly) talking about a real number divided by itself. I'm not sure what a 😄 stands for. Is it true that $0/0 = 1$? Is it true that $\infty / \infty = 1$? Etc. $\endgroup$ – Benjamin Dickman Nov 18 '14 at 20:38
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    $\begingroup$ For sake of disclosure, we had finite cows growing up, neither zero, nor infinite. $\endgroup$ – JTP - Apologise to Monica Nov 19 '14 at 0:47
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    $\begingroup$ "Fortunately" he is not yet at the level of questioning such cases as "$0/0$" or "$\infty/\infty$" so actually the 😄/😄 example was great for his intuition. I eventually covered the former of the two by the method of @JoeTaxpayer. $\endgroup$ – Therkel Nov 23 '14 at 12:43
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    $\begingroup$ @Therkel Collaborative instruction! $\endgroup$ – JPBurke Nov 23 '14 at 12:44
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It sounds to me like the student may well not understand that a fraction indicates a division.

Further, as others have already noted, this student may be performing all these computations purely formally without having any real notion of their significance. This is not at all uncommon.

I once was teaching a first-year algebra class. At one point it occurred to me, after much thrashing around, that the students perhaps didn't really understand what a fraction was. I drew a number line on the board (probably from -4 to 4) and asked where 2/3 was on the line. About half the class thought it was somewhere between 2 and 3.

These relationships -- fractions, division, points on a number line, ... are historically very deep. It took the human race a really long time to see what was going on. It's no surprise that our students have difficulty with this. What's particularly galling, though, is that so many elementary school teachers are clueless about this, and that they aren't helped much by the ed courses they take. This creates a formidable problem by the time students get to Junior High School, or even (gasp!) algebra.

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It's possible that they don't understand that x stands for a number, or that they don't understand that you are making a statement about ALL nonzero numbers, x. I'd suggest starting like this: "Let's figure out for what values of $x$, $\frac{x}{x}=1$. When $x=3,$ is $\frac{x}{x}=1$? How do we check? Well, let's see. Is $\frac{3}{3}=1$?" and lead them to a generalization.

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  • $\begingroup$ What is the key differentiating piece between this answer and the top one from CC? $\endgroup$ – Benjamin Dickman Dec 27 '14 at 11:22
  • $\begingroup$ As I mentioned in the question, he understood the most simple cases such as $\frac 3 3 $. The problem was the intuition behind it and I believe the solution to teaching was found in the answer by Chris Cunningham. $\endgroup$ – Therkel Dec 27 '14 at 18:04
  • $\begingroup$ Here's the difference, at least with my answer: my answer relates the use of "x" to the specific cases. Instead of thinking of x/x formally, we help the student think of x/x as relating to particular values of x, and makes the (near universal) quantification more explicit. If we do not make the relationship between "x" and specific numbers clear, students may think about algebra as being a set of rules for symbol manipulations that are in some way analogous to the manipulations we can do with numbers, rather than be generalizations that hold no matter what (nonzero) number is substituted. $\endgroup$ – MathTeacher Dec 28 '14 at 0:10
  • $\begingroup$ That is, the difference lies in thinking of x/x=1 as a syntactic manipulation versus thinking of it as a way of generalizing about numbers. $\endgroup$ – MathTeacher Dec 29 '14 at 22:28
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A method I would try if they understand numbers but not symbols is: Let $x\in\mathbb{R}-\{0\}$ to avoid division by zero. Now we know that $x = x^1$ and $\frac{1}{x} = x^{-1}$ (if not known explain). When we take $$ \frac{x}{x} = x^1x^{-1} = x^{1-1} = x^0 = 1 $$ where $x^1x^{-1} = x^{1-1}$ by the power rule and any nonzero real number to the zero power is $1$.

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  • $\begingroup$ This would imply they know about powers and the $x$ is exactly the problem - a bridge needs to be built between numbers and symbols before the relationship can be understood. I would also note that I believe the case of $x = 0$ should initially not be mentioned as this would only lead to confusion. Later, when they start thinking about such contradictory cases, one could limit $x$ to $x\in \mathbb R \backslash\{0\}$. $\endgroup$ – Therkel Jan 13 '15 at 17:12
  • $\begingroup$ But indeed, understanding of the concept of $x/x$ leads to a natural transition to talk about power rules! $\endgroup$ – Therkel Jan 13 '15 at 17:16
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It seems your student didn’t understand fractions, nor that the expression could be interpreted as division, and especially had trouble thinking in general terms. You asked for “a good approach to teaching this.” You got an inductive approach to generalize, answers focused on the meaning of fractions and division, and one on powers.

Here is an approach to generalization based on meaning, not inductive reasoning. It is from Davydov’s implementation of Vygotsky’s theory, references below. The approach is to start with unspecified quantities that are concrete and general, such as two line segments or two strips of wood not of equal length. Use the shorter segment as a basic unit to measure the longer segment. How many of the shorter unit fits into the longer segment? Say that the longer one is exactly 3 times the shorter. Then the answer is 3. Label the shorter line as “U” for the basic unit, and the longer as “Q” for the quantity measured. The action of measuring can be expressed as Q/U, which is measurement division: how many U’s in Q? Then for these segments we can write Q/U=3. (First graders actually get this, although it is developed in many steps.)

Then make Q the same length as U. How many U’s in Q--clearly only one. Now Q/U=1. Since Q=U in this case, we don’t need two different letters. We can write it as U/U=1. How many units in one unit? It is one unit. Now go from the general to the specific to enrich and clarify the meaning. If U is 5 then 5/5=1. There is one 5 in 5, and so on with other numbers.

I used a length of 3 at first because it is easier to understand the meaning of the measurement if the units are not the same. After establishing the measurement and division expression, then move to equal lengths. I wouldn’t think that a high school student would have a problem understanding this.

Vygotsky’s approach is to start with the general and “ascend to the specific.” The emphasis is on understanding relationships, then doing computation. For example, the additive relationships of A + B = C, C – B = A, and C – A = B are thoroughly examined in concrete situations, then symbols, before problems are given. The initial tasks are to understand the relationships. Then problems are created by making one of the three elements the unknown.

References

Moxhay, P. (2008, September). “Assessing the scientific concept of number in primary school children.” Paper presented at the International Society for Cultural and Activity Research conference, San Diego. http://lchc.ucsd.edu/MCA/Mail/xmcamail.2008_10.dir/att-0111/Moxhay_ISCAR_2008.pdf

Schmittau, J. and Morris, A. (2004). “The Development of Algebra in the Elementary Mathematics Curriculum of V.V. Davydov” in The Mathematics Educator, Vol.8, No.1, 60 – 87. http://math.nie.edu.sg/ame/matheduc/tme/tmeV8_1/Schmittau.pdf

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"he does not get it and starts guessing:

nothing? x? zero? one? "


One could proceed by elimination from the answers the student proposes.

(1) Is x/x equal to " nothing"?

Here the student probably means : " when x is divided by x, nothing remains". He is confusing the quotient with the remainder.

(2) Is it equal to x itself?

If it were equal to x, then, (x/x) - x would be 0. Is it the case?

(3) Is it equal to zero?

In order x/x to be equal to 0, x should be equal to zero, since only 0/x can be equal to 0. But that's impossible, since that would also mean " dividing by 0"!

(4) Is it 1 ?

He, one could use the fact that division "undoes" multiplication.

A little dialogue.

Teacher. Let's begin with the " multiply by x game". Which number should you begin with on the left to get the result x?

    ? ----- multiply by x ----------> x 

Student. I should begin by 1.

Teacher. Good answer : 1 is the identity element for multiplication. Now, let's play the " division game". You are on the right side and you "have" x. Wich division should you perform to " return" to 1? How could you "undo" what you've done?

   1 <--------- divide by ?  -------- x 

Student. I can't "divide by nothing" since nothing is not a number. I can't divide by 0. Should I divide by 1 ?

Teacher.- It would work in only one case, that is if x=1. Suppose x is 1000, do you hope to obtain a result of 1 when dividing 1000 by 1? As you now, in general, x/1 is not equal to 1, but to x itself!

Student. I know, I should divide by x! If I multipliy 1 by x I get x, and if I divide x by x, I find back 1, my original number!

Teacher.- Right. And you can express this truth either in the " operation language" or in the " fraction language". In the first case, you'll say that "dividing x by x gives you as result the number 1". In the other case, you'll say that, provided x is not equal to 0,

              the number x/x is the same number as 1", 

in symbols ,

         for all number x ( not equal to 0) : x/x = 1. 

Student. Isn't it strange, however, that x divided by itself gives as result precisely that special number 1 which is the identity element?

Teacher. It will look less strange to you if you notice that :

                     x/x = x * (1/x)

in other words, that x/x is x multiplied by it's multiplicative inverse! And as you know - this is the definition of a multiplicative inverse - if I multiply a number A by it's inverse 1/A, the result is ( by definition I repeat) ....

Student .- .... the identity element ( for multiplication) : 1 !

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