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To find a basis for the row space of a matrix, perform elementary row operations to get to reduced row-echelon form. The nonzero rows of the row-reduced matrix are a basis for the row space of the ...

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It might be enough to show an example or two of what can go wrong with the logic of the first solution: $\lim_{x\to\infty} \frac{e^x}{x^2}=\frac{\infty}{\infty^2}=\frac{1}{\infty}=0$ But of course ...

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I read the left-hand side as 11 minutes and 40 seconds minus 15 minutes. (Analogous to 11.4 - 15). So I expected the right-hand side to be negative. I would not write this, especially as a teacher. I ...

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Ask him whether the probability of winning is the same if you bought 1000 tickets rather than one ticket. Or, imagine a lottery with 100 tickets, of which only one was a winner. If 100 different ...

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Our MAA section (North Central) has an annual team math competition for undergraduates. It has proven more attractive to the students than the Putnams (which we haven't tried for many years). The ...

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My sense is that often authors are encouraged (required?) by publishers to make books very 'complete'. This means the book has all the topics that any instructor would be likely to want for a subject. ...

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I am someone (in a country), and I teach it in certain (college) classes. In a discrete math class I teach it and then use the fact that if you multiply $n\times 1$ with this method (halving $n$ ...

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How about : rescaling the fraction. This could refer to going in either direction.

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A compromise approach could be to give a problem (with parts), as you might on a guided worksheet. Such as: A function $f$ has domain $(2,4)$. We define $g$ by $g(x) = f(x-2)$. a) Is $g(3)$ ...

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Another example of this sort of issue: I have often seen students write things like $\frac84$ when they mean $4\mid 8$ (i.e., $4$ divides $8$.) It takes a while (for some) to see that the former has ...

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How about this: I made $\$12$per hour for$7$hours. I then bought as many books as I could for$\15 each. How much money did I have left? This would illustrate $7\cdot 12\equiv ? \pmod {15}$.

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In the game of Set, https://www.setgame.com/set, you can define a binary operation on the cards by Card1 * Card2 gives the card that completes the 'set'. (In the case that the two cards are equal, ...

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