Post 4 of Wally's Math Books: Explore Like a Mathematican

About a week ago I listened to a podcast called "Can we help kids learn to love math?" which featured math educators Christopher Danielson (@trianglemancsd) and Sara VanDerWerf (@saravdwerf). One of the key takeaways from this is that math is NOT about calculating fast. In fact, we destroy kids views of themselves as mathematicians when we tell them that being 5 seconds slower at a calculation is bad. The truth is math is about exploring and playing - things kids enjoy. Math is about noticing, describing and generalizing.  These things can happen naturally when students are given time to play with math.   

So what does playing with math have to do with Wally's math books?  If you look at my grandfather's college math textbooks from the 1920's, you'd think that math is all about solving things in one way, using theorems and performing calculations.  However, I did find glimpses of what mathematicians actually do in this book - the making observations, looking for patterns and drawing conclusions part of mathematics. Here is one such question in the chapter called "General Theory of Equations":

                    

I graphed these three functions on Desmos.  It was easy to see that the functions were just vertical transformations of each other.  However, I was curious about what happens with the x-intercepts.  So, I created a table which included these functions and a few others

Wow!  The sum of the x-intercepts is 3.  This leads to a whole list of questions.

• Why is the sum of the roots 3? 
• Is it related to coefficient of the quadratic term, which is -3?  
• What would happen if I changed that coefficient? 
• If I changed the coefficient to -4, would the sum of the x-intercepts be 4? 
• What if I added a linear term to the function? How would the sum change?
• What about the product of the x-intecepts? How is it related to the coefficients of the cubic polyonomial?
• How is the sum and product of the roots impacted if the polynomial is an even degree?

Noticing, wondering, generalizing...thanks to Wally and his math book for giving me something new to learn about polynomials and giving me an opportunity to think like a mathematician.

Post 3 of Wally's Math Books: 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.

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: https://www.desmos.com/calculator/r6p2hpdplf

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.

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.