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30 votes

Why is absolute value difficult?

(My answer is just a guess and not based on any formal research.) I suspect the absolute value function may be difficult to understand because it involves "negative numbers that aren't negative." ...
JRN's user avatar
  • 10.9k
19 votes

Why is absolute value difficult?

My experience is that weak students latch onto absolute value of a number is always positive. They are fine working with constants. When you introduce a variable, it all falls apart. To these ...
Amy B's user avatar
  • 8,017
12 votes

Why is absolute value difficult?

In my experience, this is the first really clear example that students experience of dissonance between how something looks and what it is. One of the most common errors I see regarding absolute value ...
Reese Johnston's user avatar
10 votes

Why is absolute value difficult?

I am writing this based on pure observation (e.g., entering year four of teaching this topic to secondary school students, and having co-taught a minicourse for teachers on absolute value functions$^\...
Benjamin Dickman's user avatar
8 votes

Is there a standard convention for interpreting ambiguous absolute value expressions?

I don't know about standards, but I read these things using a left to right, greedy algorithm. More specifically, the bars are like parentheses but you don't automatically know if they are opening or ...
Adam's user avatar
  • 5,873
8 votes

Is there a standard convention for interpreting ambiguous absolute value expressions?

If I want to have nested absolute-value expressions, I would use different sizes $$ \big|x + 2|x + 3|x + 4\big|, $$ with variations possible $$ \bigg|x + 2|x + 3|x + 4\bigg|. $$
Gerald Edgar's user avatar
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5 votes

Why is absolute value difficult?

Absolute value is difficult for students because they have difficulty parsing and simplifying logical statements. Some of the results of working with absolute value statements seem to actually hide ...
Mark B's user avatar
  • 133
5 votes

How can one lone picture prove the Triangle Inequality, $|x−y|≤|x|+|y|$, $|x|−|y|≤|x−y|$, and the Reverse Triangle Inequality?

First of all, you want them them to "get it right" on a test, rather than be able to use this to more deeply understand vectors. In fact, it seems to be about memorizing this, which is ...
Sue VanHattum's user avatar
  • 21k
4 votes

Is there a standard convention for interpreting ambiguous absolute value expressions?

As I was looking at your expression, something just seemed typographically off, and then I realized that it was the missing padding around the bars that you see when mathematics is well-typeset. This ...
Kyle Miller's user avatar
4 votes

How can one lone picture prove the Triangle Inequality, $|x−y|≤|x|+|y|$, $|x|−|y|≤|x−y|$, and the Reverse Triangle Inequality?

Please remember these 4 inequalities. Save these pictures as screen savers or wallpapers, hang them by your toilet and bedroom, do what you got to do. I always showcase separate pictures . . . I ...
Justin Skycak's user avatar
4 votes

Why is absolute value difficult?

It is difficult because there are many things implicitly done behind the scenes: Function definition with case analysis usually for the first time. Solving problems, among which are equations and ...
Physics Novice's user avatar
3 votes

Why is absolute value difficult?

This answer is concerned with real numbers only. Adding to Dave Renfro's comment about absolute-value manipulations being—or at least feeling—different from usual algebraic manipulations, and Mark B's ...
ryang's user avatar
  • 1,832
2 votes
Accepted

How can one lone picture prove the Triangle Inequality, $|x−y|≤|x|+|y|$, $|x|−|y|≤|x−y|$, and the Reverse Triangle Inequality?

Another possibility is to show a sequence of diagrams explaining how the inequalities relate to one another. The top left diagram shows the triangle inequality, $\lvert \mathbf{a}+\mathbf{b}\rvert \...
Will Orrick's user avatar
  • 1,122
1 vote

How can one lone picture prove the Triangle Inequality, $|x−y|≤|x|+|y|$, $|x|−|y|≤|x−y|$, and the Reverse Triangle Inequality?

You are telling sixteen-year olds that: $$|x - y| \le |x| + |y|$$ ... and you call that "triangle inequality"? ... and your teenage students don't get that? Well, I'm a mathematician and I ...
Dominique's user avatar
  • 2,175
1 vote

Is there a pre-calculus introduction to the formal definition of a limit?

Precalc courses differ a lot, so it's hard to say something is included or not. After all there was a time when there didn't even exist a precalc course (different from a strong algebra 2, trig, and ...
guest's user avatar
  • 134
1 vote

Is there a pre-calculus introduction to the formal definition of a limit?

Perhaps what would best prepare a student for the formal $\epsilon{-}\delta$ definition of a limit is an animation which shows both $\epsilon$ and $\delta$ approaching $0$, $\epsilon \neq \delta$, ...
Joseph O'Rourke's user avatar
1 vote

Why is absolute value difficult?

IMHO the main issue is that the absolute value doesn't satisfy any simple algebraic rules, so you cannot simplify the expressions including it. The computation itself is no more difficult than the ...
fedja's user avatar
  • 3,939

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