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I assume your students have seen Linear Algebra. Remember in Linear Algebra how you sometimes have to solve $Ax = b$ for a matrix $A$ with more columns than rows? You usually get free variables, right? So after reducing to echelon form you have something like: \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \...

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I think that it makes sense to introduce continuity in the same lesson that you introduce limits. Here is a sketch of a lesson plan: Give them this link https://www.desmos.com/calculator/rlu2zgcjyf Group work: Is $f(2)$ defined or undefined? As $t$ approaches $2$, what are the values of $f(t)$ approaching? Make a table of values for $t = 1.9, 1.99, 1.999$ ...

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Equivalence classes provide a tool for describing information. If you have an object $x$ and an equivalence relation $R$, one piece of information about $x$ is "which $R$-equivalence class is it in". What's more, any piece of information can be described in this way by constructing an equivalence relation where two things are equal if this ...

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