Random Rendevous with Desmos

Over a year ago I heard statistics professor Allan Rossman speak during a virtual session. The problem he shared was easy to understand and potentially challenging to solve. 

Here's the scenario:

Two friends agree to meet for lunch.  They agree to wait for 15 minutes for the other person to arrive. If they don't arrive in that 15 minute window, then they leave. The friends arrive at a random time between noon and 1 PM.  What is the probability they will see each other?

Allan demonstrated how to solve this problem using a simulation in R and I decided to try creating a simulation for the problem using Desmos.  I presented my Desmos version of this solution during a 15 minute Epsilon Talk at Moravian University during the spring semester.  This blog will take you through that talk.

Let's consider the following times.  Do they work or not?

To help us think about all the different random rendevous pairings, we will use the following definition. 

Let (A, B) represent the minutes after noon that Person A arrives and the minutes after noon Person B arrives.

Example: Person A arrives at 12:15 PM and Person B arrives at 12:56 PM. This would be represented as (15, 56).

Here's an image that shows many ordered pairs like (15, 56). Which pairs correspond to meeting for lunch?

Remember that there is a 15 minute window were each person will wait for the other person. This means 

and we can color code our Desmos graph, where red dots mean that the two friends met and blue dots means the friends did not meet. Is there a pattern that emerges?

I invite you to go to this Desmos calculator page and use the slider to investigate adding more random ordered pairs.  What shape is formed by the red dots?  

Other Desmos simulations can be seen here and here.  The first simulation creates a histogram with the width of each bar being 15 minutes. The second simulation uses a uniform distribution and a ticker and was created by Andrew Knauft. 

Of couse any good question leads to more questions...here are some others that I invite you to explore:

• How long should the friends wait if they want the probability of meeting for lunch to be 50%?

• What if they wait different amounts of time?

• What if they arrive randomly in a 30 minute window of time?

• What if more than 2 people want to meet for lunch?

Please share your answers to these questions or your own random rendevous questions.

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.