AQOTW

Order of Operations: Fractions, Evaluating, and Simplifying

(An archive problem of the week) Last time we looked at the subtle distinction between the order of operations, which defines the meaning of an expression, and properties that allow us to do something other than what an expression literally says. Here I want to look at one longer discussion that brings out these issues …

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Order of Operations: Trigonometric Functions

(An archive question of the week) Last time we looked at some details that are rarely mentioned in stating the conventions for interpreting algebraic expressions. I couldn’t fit a discussion of the most complicated case: trigonometric functions, which when written without parentheses, as they traditionally have been, can raise several issues. (Much of the same …

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An Introduction to Trigonometry

(An archive question of the week) While I’m showing some recent explanations of basic trigonometry techniques, this is a good time to look at an even more basic explanation of the essentials of the subject for a beginner. Right triangle trigonometry Here is the question, from 2001: Trigonometry in a Nutshell I’m in 8th grade …

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Equations with Fractions: Three Ways to Solve Them

Since we just looked at a complicated rational inequality, let’s look at some simpler rational equations, first a linear equation with fractions, and then truly rational equations, in which the variable(s) appear in the denominator. This discussion dealt with a common confusion I’ve seen in students. The problem The question came from Fairooz in 2017: …

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How Many Different Pizzas?

(An archive question of the week) We’ve been looking at examples of extended discussions with students about various kinds of problems. Here, we have one (not from a student) that led to some good thinking about combinatorics – the techniques of counting the ways something can happen. The problem: Triple toppings Here’s the question, from …

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Too Many Variables?

(An archive question of the week) Students often struggle with solving an equation with several variables, for one of those variables. This is also called “solving a formula”, or a “literal equation”; or “making one variable the subject”. Learning to use variables instead of just numbers (as we looked at last week) is the first …

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