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1

Modular arythmetic works just like a clock I've seen several students expect "half-hours" making an appearance in $\mathbb{Z}_{12}$. This leads them to be confused about statements like $5^{-1}=5$ rather than $5^{-1}=0:12$

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A point (say, in $\mathbb{R}^n$) is a vector. Vectors and points are really no different. They are both $n$-tuples in $\mathbb{R}^n$. The difference between two points (in $\mathbb{R}^n$) is a vector, but a vector has no fixed position. Points are positions in space. Vectors are displacements. It makes no sense to add two points, but it does make sense to ...

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For a much lower-level topic, consider explaining to beginning algebra students why "like terms" can be combined. On a few occasions, I have resorted to reasoning with students that adding algebraic expressions is like adding quantities with units. [Our curriculum begins with units and geometry before algebra, so this is usually safe ground in my class.] If ...

3

I shall post my humble and incomplete list of bad explanations I've given or heard over the years: A function is continuous if you can draw its graph without lifting your pencil As you mentioned this is bad, but, depending on the level of the student, it can be a reason for big or small misunderstandings. At a high school level, this simply ignores the ...

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It is possible for an expression to be undefined. If you work in $\mathbb{N}$, you can consider $2 - 3$ to be undefined. We just have meaningless expressions. Similarly when working in $\mathbb{R}$, we can also consider the expression $\frac{1}{0}$ to be a meaningless expression. $\frac{1}{0}$ is considred undefined because $\frac{1}{0}$ means the number ...

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