Monthly Archives: February 2020

The Za’irajah and Mathematics

The subtext of this essay might be “word problems,” since the stream of thoughts that led to the za’irajah (zairja) began with a paper I read, while searching for potential problems for this website, on the history of word problems in high school texts in algebra in the 20th and 21st centuries. The following statement by Lorenat caught my attention:

“The newer characteristics of how word problems are treated in Long’s text [2016] … include adding sympathetic commentary about fear of word problems. ‘And of course, there are those dreaded “Word Problems,” but I’ve solved them all for you, so they’re painless.’

[Lorenat continues,] A more extreme example of this is exhibited in the word problem commentary of Michael W. Kelley’s The Idiot’s Complete Guide to Algebra: Second Edition from 2007, in which he describes word problems as ‘a necessary evil of algebra, jammed in there to show you that you can use algebra in “real life.”’ However, Kelley makes no attempt to write ‘real life’ word problems, and criticizes the uselessness of the word problems he does include.”

This essay is an attempt to rebut such negative views of solving word problems by placing the activity in a more favorable historical context.

See the Za’irajah and Mathematics

Railroad Tie Problem

This is a surprisingly challenging puzzle from the Mathematics 2020 calendar.

“The sketch is of equally spaced railroad ties drawn in a one point perspective. Two of the ties are perceived to the eye to be 25 feet and 20 feet respectively. What is the perceived length x of the third tie?”

Even though the ties are equally-spaced and of equal length in reality, from the point of view of perspective they are successively closer together and diminishing in length. The trick is to figure out what that compression factor is. I had to review my post on the Perspective Map to get some clues.

See the Railroad Tie Problem

Perpetual Meetings Problem

The following problem from Five Hundred Mathematical Challenges was a challenge indeed, even though it appeared to be a standard travel puzzle.

“Problem 118. Andy leaves at noon and drives at constant speed back and forth from town A to town B. Bob also leaves at noon, driving at 40 km per hour back and forth from town B to town A on the same highway as Andy. Andy arrives at town B twenty minutes after first passing Bob, whereas Bob arrives at town A forty-five minutes after first passing Andy. At what time do Any and Bob pass each other for the nth time?”

Answer.

See the Perpetual Meetings Problem for solutions.

Pinwheel Area Problem

Here is another engaging problem from Presh Talwalkar.

“___________Triangle Area 1984 AIME
Point P is in the interior of triangle ABC, and the lines through P are parallel to the sides of ABC. The three triangles shown in the diagram have areas of 4, 9, and 49. What is the area of triangle ABC?”

Answer.

See the Pinwheel Area Problem. for solutions.