Friday, January 7, 2022

Post 3 of Wally's Math Book: The Solution to the Grapefruit Problem

In my previous post I noted that one of the Wally's textbooks was explicity written to cover all of what was on the New York Regents exams, which still exist today - although the ones for January 2022 have been cancelled due to COVID-19.  What about these exams is so enduring that thay have lasted since 1865?  The exams have been cancelled multiple times more than once during the pandemic and that makes me wonder.  Are these exams really all that important? And if not, why administer them at all?  But I digress.

Here is the problem I had posted previously.  Did you try to solve it or give it to some students to try to solve?

Since I am a math teacher, I decided to approach this from an algebraic perspective. But then I got an equation that requires the quadratic formula and so I switched to using Desmos.  After you students try to solve it, see if they can follow my paper/tech combo solution.

In the next post, we will take a look at logarithms. They had a very practical use before the electronic calculator and I even have my grandfather's slide rule.

Thursday, December 16, 2021

Post 2 of Wallly's Math Books: Fundamental Laws and Operations

This is the second post of a series of blogs devoted to my grandfather Wally's college freshmen math textbooks. One of the books Wally used was called Advanced Algebra. It was published in 1925 and in the preface it states the book is designed for students who have had one or two years of college algebra.  In addition, it states "The syllabi of the College Entrance Board, of the Regents of the State of New York, and of practically all states have been covered." This makes me wonder if there are college math texts today that take the New York Regents or Common Core Standards into account when designing their textbook.

In the previous blog post, I encourgaed people to try the following problem.  In addition to posting on twitter, I linked the blog post to the "Higher Education Learning Collective" Facebook group and was told by some that this problem was too easy to be a college level problem.  Here was that problem.

Problem 1: Prove that "The sum of the squares of any two unequal real numbers is greater than twice their product.  That is if a ≠ b, a2 + b2 > 2ab."  

Here is essentially the algebraic proof as given in the textbook: 

Given: a ≠ b, we know that a - b ≠ 0.  Squaring both sides gives  a2 - 2ab+ b2 > 0 and that means that a2 + b2  > 2ab.

Here is a way to visually see that the sum of the areas of two squares is more than double the area of a rectangle formed by the sides of the squares by using Desmos - a much more dynamic and visually appealing proof.

And here's a link to the original Desmos page that I used to create the video above:

grapefruit Problem 2: This problem comes from pg. 28 of the Chapter 1 Fundamental Laws and Operations.

Although it doesn't appear to have been assigned to my grandfather and his classmates, it seemed like a interesting problem to solve. In fact, it is problem that I could see myself giving to my MATH 120 (Math for Teaching) students on the first day of class just to see what problem solving strategies they might use.

A man bought a certain number of grapefruit for $1.04. After throwing away 4 bad ones, he sold the others for 6 cents apiece more than he paid for them and made a profit of 22 cents. How many did he buy, and what did he pay for each?

How would you solve this problem?  Are there any generalizations you can draw?  Feel free to post in the comments of this blog post.  We'll look at at least two approaches to this problem in my next post.

Sunday, November 28, 2021

Post 1 of Wally's Math Books: The Syllabus + Problem 1

As I opened up my grandfather's math books - Advanced Algebra by Edgerton and Carpenter, published in 1925 and Plane Analytic Geometry by Barnett, published in 1927 - I saw a piece of paper folded up.  I carefully opened it and realized it was the course syllabus! How exciting to see how a syllabus from the fall of 1927 compares to the current syllabi I write for the math classes I teach.  Before you read my observations about this document, I encourage you to look at it yourself.  Zoom in.  What do you notice? What do you wonder?  Feel free to add a comment to this blog post.

Math 22 Syllabus from 1927

What I Noticed 

The paragraph at the top of the paper contains many instructions found in a typical syllabus. 

  1. Time Spent Outside of Class - It says that each assignment "merits two hours of study and review". My syllabus says "You are expected to spend at least 7 hours outside of class working on problems or studying for this class." 
  2. Number of Classes Per Week - Although I don't see how long a class is on this syllabus, the class meets 4 times each week.  This would translate into 8 hours of work spent outside of class.  My class meets 3 times each week for 70 minutes each class period.  So, it is possible that the amount of in class time may be similar.
  3. Ask Questions and Get Help - It is clearly stated that the instructor Professor Waltz wants the students to ask questions during class and outside of class.  There is no mention of specific office hours.  Maybe office hours were a construct that became standard after 1927. 
  4. Saturday Help Sessions - There is time and a location for help.  Saturday at 9 AM would not go over well for most students. But then again, it is a time when all of them would be available.  We currently offer virtual help sessions 11 hours a week and every day of the week. except Saturday!

What I Wondered
  1. What problems should be done? The list of problems was outlined, but what did 5n, 2n mean?  As I looked in Wally's math books, I noticed that he had checkmarks beside certain problems.  Every 3rd problem or every 5th problem. So, this notation was to show multiples of a given value.
  2. Assessments Where there quizzes or exams?  It says "Final Tests" on class meeting #62.  Since it says "an average of A or B will exempt from final tests", there must be something that is being graded.  But what?
  3. Where are the problems found? There are two books listed and it looks like Plane Analytic Geometry book is used starting on October 31.  However, it is not clear that Advanced Algebra is the first book.  And does October 10 - 97(5, 8, 10) mean page number 97 and problems 5, 8 and 10 was the assignment on October 10th?  Maybe...but neither book has a page 97 with those problem numbers.
  4. What is up with the margins? What is the arithmetic that Wally did on the side?  It looks like these may be his grades in the class and he is averaging them.  Perhaps he is trying to be sure he is exempted from the "Final Tests".
What did you notice and wonder about the syllabus?

Problem 1: This is not a problem that was assigned, but whas listed as a theorem on page 70 of Advanced Algebra.  It states "The sum of the squares of any two unequal real numbers is greater than twice their product.  That is if a ≠ b, a2 + b2 > 2ab."  Feel free to create your own proof and upload it in the comments or on twitter.  Make it an algebraic proof or a proof without words.  I have already started to work on a way to visualize this with Desmos and will share my solution in the next post.