*“His life was gentle, and the elements mixed so well in him that Nature might stand up and say to all the world, ‘This was a man.’”*

* — Julius Caesar, *William Shakespeare

This month, Maclear “Mac” Jacoby, longtime Middle School Headmaster at my alma mater Landon __School__ in Bethesda, passed away from Covid-19. He was also Landon’s longtime tennis coach and a terrific geometry and algebra teacher.

On Monday, The Washington Post ran an appreciation of Mr. Jacoby’s life and I’m quoted a few times:

“Mac’s classes were unique,” said Peter Arnold, who graduated in 1982 from Bethesda’s Landon School, the all-boys prep school in suburban Maryland where Jacoby taught math for more than two decades. “You couldn’t wait to see how he’d use a little craziness to teach the finer points of algebra and geometry.”

Jacoby beguiled rowdy seventh- and eighth-grade boys with tales of his “dear Aunt Sally” — a mnemonic device the teacher used to remind his students to “divide, add and subtract” only after they multiplied the numbers in an equation. It’s been 45 years since Arnold sat in Jacoby’s chalk-covered classroom, but he still remembers the tricks his teacher taught him, including a shortcut for squaring two-digit numbers in a matter of seconds.

Incidentally, as a tribute to Mr. Jacoby (yes, even 38 years after graduation, he’s still “Mr. Jacoby”), here’s the explanation referenced in the 4th paragraph about squaring numbers ending in “5″: When squaring a number ending in 5, write the number “25” on the right of the answer space. The number(s) on the left are the product of the original number’s other digit(s) multiplied by 1 more than that number.

Therefore, to square 65, you’d write “25” on the right side of the answer space and then “42″ (i.e, 6×7) to the left. Viola! 65 squared is 4,225.

I shared this solution this week with my longtime Dawson & Associates colleague Gen. (ret) Rick Stevens. Not surprisingly for a West Point grad who became Deputy Commanding General of the U.S. Army Corps of Engineers, Rick provided the “man behind the curtain” explanation. With appreciation to Rick, here you go:

Any whole number that ends in 5 is of the form A + 5, and A is a multiple of 10. Since A is a multiple of 10, we can write A = 10b, where b is now some whole number. So, our number is of the form 10b + 5. Now, let’s square it.

(10b + 5)(10b + 5) and we use the distributive property to multiply this out.

(10b + 5)(10b + 5) = 100b2 + 50b + 50b + 25 = 100b2 + 100b + 25

Now, notice those 100b’s there? We can gather that as a common factor for the first two terms:

= 100 b (b + 1) + 25

This is now essentially in the form that the trick is using. The trick says to take b, or the number formed by the digits in front of the 5. So we take b, multiply it by (b + 1) which is the next number, and also by 100, and lastly add 25.

Now, b × (b+1) is the part of the trick where you multiply the digits in front of the 5 by the next number. To “tag” 25 to those digits means you add 25 only after having multiplied the number by 100 so that it would end in “00”. Once it ends in “00” you can add 25 (or any two-digit number) and it is the same as “tagging” 25 to the digits without the “00”.

Rest In Peace, Mr. Jacoby. I hope someday we’ll see each other again.

UPDATE: Rick just shared that he used http://homeschoolmath.blogspot.com (including a cut/paste of elements of the proof) to refresh his memory on why Mr. Jacoby’s trick works.