A recent question, from Anindita, touched on the relationship of functions, relations, and rules. I referred to several answers we’ve given, which I’d long planned to put into a post (or two). This is it! We’ll start with a set of questions about what functions are.
Last week, we looked at problems involving some number of people making some number of things in some amount of time. In a classic twist on this problem, we’ll now examine several variants starting with “If a hen and a half can lay an egg and a half …”. Can we make sense of half-eggs …
A popular kind of word problem tells us how many people (or cats, or hens, …) it takes to make some number of houses (or kill some number of mice, or lay some number of eggs) in some amount of time, and then asks us to fill in one of the blanks for a different …
How Many A’s Can Make This Many B’s in This Much Time? Read More »
(A new question of the week) I find it interesting to observe the process of problem-solving, particularly for proofs: how we discover a solution initially, and then how we turn that into a final answer. Sometimes we can see the main idea in a flash, but the process of writing it as a formal proof …
One of the recent discussions I showed last week dealt with the meaning of length, and I promised more about that. Here we will look at some older questions about the ambiguity of words like length, width, depth, and height.
(A new question of the week) Several recent questions involved details about definitions of geometrical objects, so I thought I’d group them together, because each is relatively short. We’ll be looking at the definitions of triangles (do we need to say “exactly three sides”?), rectangles (did Euclid use an exclusive definition?), and circles (can the …
Clarifying Definitions: Triangle, Rectangle, Circle Read More »
(A new question of the week) I intended to fit three problems into last week’s post, but the third was too interesting to shorten, so I’m posting it separately. The problem itself is not hard, but in looking for a more direct solution, we extend it, discovering (through geometry software) more general facts, which lead …
(A new question of the week) Several interesting geometry problems about triangles and circles came in recently. We’ll look at two today, and a third next week.
Last time we looked at basic methods for finding the prime factorization of a number. Here we will look at some special techniques for large numbers, demonstrating them for not-too-large numbers. This takes us a step beyond previous tests that told us whether a number was composite, without actually factoring them.
I’ll close this series on prime numbers by looking at how to find the prime factorization of a number, starting with the most basic ideas applicable to relatively small numbers, and then (next week) looking at some advanced methods for larger numbers.