Parentheses and the Associative and Distributive Properties

(A new question of the week) This week, we’ll look at two recent questions about how parentheses (brackets) are used, how they relate to the properties we use in algebra that let us add or drop them, and the related concept of factoring a polynomial. They are examples of how student questions can touch on …

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Probability: Cards vs Dice

(A new question of the week) A couple recent questions involved related subtleties in probability and combinatorics. Both were about apparent conflicts between similar problems involving cards and dice.

Non-homogeneous Recurrence Relations

(A new question of the week) A recent question asked us to find errors in solving recurrence relations by the method of undetermined coefficients. We’ll see several things that can go wrong, and correct some misunderstandings.

Two Integration Puzzlers

Two recent questions (that came to us within two hours) dealt with apparent contradictions in integration. The first seems to give a result of zero that is clearly wrong; the second seems to give two different results for the same integral.

Help with Factoring: Trinomials

(A new question of the week) I recently had a pleasant discussion of factoring, with the kind of student for whom I returned to teaching: one who has been away from math for a while, and with greater maturity has the determination to succeed. We’ll see several examples of the “ac-grouping” method of factoring a …

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Implicit Differentiation: Explanation, Examples, and a Surprise

In response to a recent request for information about implicit differentiation (hi, Brian!), let’s take a look at that topic. It happens to be distantly related to Friday’s topic, which was about implicitly defined curves. We’ll start with a thorough explanation, and then look at several specific examples, capping it off with a weird one.