(A new problem of the week) Last week I mentioned “non-routine problems” in connection with the idea of “guessing” at a method. Let’s look at a recent discussion in which the same issues came up. How do you approach a problem when you have no idea where to start? We’ll consider some interesting implications for …
Last time we looked at applying Heron’s formula to problems about the area of a triangle, where knowing the side lengths is enough to determine the area; there was a passing mention of the fact that more is needed for quadrilaterals. We’ll start here with a repeat of that idea, and then look at several …
Last time we looked at a very useful formula for finding the area of any triangle, given only the lengths of its sides. Today I want to look at several problems in which the formula has been used, some of them surprising.
The next few posts will examine aspects of logic, both symbolic logic, and how we talk about theorems in general. We’ll start here with issues in interpreting the wording of logic, and some of the semantic difficulties we face. English isn’t logical. (Well, I suppose humans in general aren’t logical.)
I’ll close out our look at transformations of functions with some trigonometric graphs. These are the best example of combined transformations, and involve some special tricks as well. We’ll start with an early question that gives an overview of the process, then focus in on important details.
(An archive question of the week) We’ve looked at the basic transformations of a function and how they affect its graph, then at how they combine, and then how they can interact with specific functions. Now let’s look at one problem from beginning to end, looking at a graph and finding the function that goes …
Last time we looked at questions about how to shift, stretch, or flip a graph by changing the equation of a function. All our examples involved only a single transformation. Now we can look at cases where two or more transformations are combined. As we do this, we will develop a deeper understanding of how …
Combining Function Transformations: Order Matters Read More »
(A new question of the week) I want to look at a question that came in recently that is, in one sense, very simple, but at the same time is quite challenging. It was given to a 12-year-old whose father asked us about it, and requires some skill in thinking about non-routine problems.
We have been looking at ways to count possibilities (combinatorics), including a couple ways to model a problem using blanks to fill in. Today, we’ll consider a special model called Stars and Bars, which can be particularly useful in certain problems, and yields a couple useful formulas. (I only remember the method, not the formulas.)
(A new question of the week) We have been looking at some combinatorics questions, both easy and challenging. Some questions have come to us in recent weeks that can illustrate how to think your way through relatively difficult problems, including catching errors and interpreting a textbook’s solutions. We’ll see yet again that there are usually …
Permutations and Combinations: Undercounts and Overcounts Read More »