Having just discussed why we use compass and straightedge in geometry, let’s flip that around and look at a common question at the more elementary level: How do you use a ruler to measure or draw a line of a given length? The usual issue here is working with the fractional markings on an inch …
Some time ago I looked at questions about trisecting an angle by compass and straightedge, which entailed discussing the rules for such constructions. We left open another common question: Why are such constructions important, and why do we use those particular tools? This probably isn’t explained as often as it should be. Why does it …
To finish up this long series on the order of operations, I want to look at where the “rules” came from, which will also demonstrate why some aspects are not fully agreed upon, finishing up the discussion from last time.
I want to close this series with a topic that arises constantly, both in classrooms and on social media: How do you evaluate an expression like or , where the multiplication is indicated without a specific symbol? There are several reasons one might want to interpret this differently than the rule we’ve discussed, that multiplication …
(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 …
Order of Operations: Fractions, Evaluating, and Simplifying Read More »
Some questions we have been asked about the order of operations go beyond the what and why, pondering the relationship of the conventions both to theoretical matters (properties of operations) and to practical matters (evaluating and simplifying expressions). We will see here an important distinction between meaning and processes.
Last time we looked at some questions about why we need rules for Order of Operations at all, with some hints in the answers as to why the rules we use make sense. This time I want to survey some deeper explanations.
Having looked at what the order of operations convention means, another common question is, why is it what it is? We’ll look at some basic ideas here, focusing on why we need a convention at all, and why the one we have makes sense; then next time we’ll dig in a little deeper, examining some …
(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 …
The basic statement of the order of operations covers the five main operations (exponents, multiplication, division, addition, and subtraction). But what about other operations like square roots? How about trigonometric functions? And are operations at the same level always carried out left to right? Here are some questions about the details we don’t often mention.