A version of this post first appeared at firstname.lastname@example.org on 8/12/13.
This morning when I got to work, I noticed that “1912 eighth grade exam” was trending on Yahoo!:
They show a recently recovered test for 8th grade graduation from 1912. (You can see a picture of the entire exam in the Washington Post, I’m showing you the arithmetic part of the exam below).
It looks like there were a variety of subjects 8th grade students were supposed to be familiar with, but I was really interested in question 8 in the arithmetic section.
How long a rope is required to reach from the top of a building 40 ft high to the ground, 30 ft from the base of the building?
I was interested in this question because I just happen to have in these in my office:
That’d be my great grandma’s Arithmetic, Geometry, and Algebra textbooks from when she was in junior high and high school. And, that’d be a picture of her work doing polynomial division, as well as her perfect attendance for the month of December 1917. The copyright on the books range from 1879 – 1911, so the material in the books was written right around the time that this eighth grade graduation exam was given.
Can we also just recognize how much of a mathematical rockstar my great grandma must have been!?! (My understanding is that she was also a great oatmeal chocolate chip cookie and Chex Mix maker . . . all of the important things in life; math and food)
Anyway, I was interested in looking through her textbooks, because I was interested in trying to figure out how students in 1912 would have been expected to solve this problem. First, if we can agree that the situation described creates a right triangle, then I think we can consider that this problem could have been answered one of three ways; by recognizing and memorizing the Pythagorean Triples, using the Pythagorean Theorem, and by using Trigonometry.
Now, if you have the Pythagorean Triples memorized, you know that the smallest right triangle (in terms of perimeter), with all side lengths being integers is a 3-4-5 right triangle. Then, the triangle formed in this situation is a 30-40-50 right triangle (the smallest P. Triple triangle, times 10), so the rope must be at least 50 ft long.
But, her textbooks didn’t make any mention of the P. Triples (I was a little surprised, I thought math was all about looking things up in tables back then). But, her Plane Geometry textbook did have this:
leading me to believe that these test-writers probably expected students to use the Pythagorean Theorem, also giving a minimum rope length of 50 ft.
The last way to address this problem would be by using trigonometry, but I doubt this was the case. I didn’t find trigonometry in any of her textbooks. I know people knew about trigonometry in 1912, but my guess is that branch of mathematics was reserved for the students in upper level mathematics classes, and that, that particular method of solving probably wasn’t expected on the 8th grade exit exam in 1912 (or now, for that matter).
Also, trying to answer this question using trigonometry certainly isn’t incorrect, but it makes this question a multi-step one. However, just in case you’re interested here’s the way we could have used trigonometry to help answer our question:
tan ß=30/40; ß≈0.6435; sin(0.6435)=40/x; then x=50.
Not that you could find them, but if you’re interested here are the citations for the textbooks I consulted:
Hart and Feldman (1911). Plane Geometry. American Book Company. NY, NY.
Milne. (1911). First Year Algebra. American Book Company. NY, NY.
Ray. (1856). The Principles of Arithmetic, Analyzed and Practically Applied for Advanced Students. Van Antwerp, Bragg & Co. Cincinnati.
Wells. (1909). Essentials of Algebra for Secondary Schools. D.C. & Co Publishers. Boston.