I’m sure you know someone who’s entered a race whether it be, running, cycling, swimming, skating or skiing. A friend. Family member. A student. Yourself! Maybe you watch NASCAR, walk with a friend or commute to work?  If so, Then this post is for you! I’m going to discuss SMP #4: Modeling with Mathematics which requires the math to have a real world connection.  Therefore, I’m going show how mathematics is connected to my upcoming half marathon – which is 1 week away. Although I’ll be focusing on a particular situation, the concept discussed can be applied to a variety of races as well as your commute to work or school.

In the spirit of running races, I would like to dedicate this post to my dad, T. David Raftery and my brother, Davey. On July 4, 1975 they entered the Merrymount Road Race in Quincy MA.  My brother placed first overall while my dad placed first in his age group – A great day for the Raftery men.

There’s a famous quote that states: A picture represents a thousand words.  In math, equations and graphs are our pictures.  Surpisingly, the equation, d = rt or distance = rate(time), can paint a picture of the 2015 Tri-City Half-Marathon in Carlsbad, CA

In one week, I will stand at the starting line. When the gun goes off, I will travel 13.1 miles. That’s all I know for sure.  I don’t know what my time will be or what my average speed will be until I cross the finish line.  Mathematically, this piece of information is represented using the following equation.

Distance     =      Rate                x                  Time      

13.1 miles   =   my speed (mph)      x            my  time (hrs)                           unknown, r                         unknown, t

     13.1        =        6 mph           x            2.183 hrs                                            (See the graph below)

The story of the Tri-City Half Marathon is summed up with the equation:  13.1 = rt.  

The equation, 13.1 = rt, represents all the possibilities that could happen on race day.  It represents all the possible speeds (rate) that I could run which would determine my final time (time).

Those possibilities are represented in the following graph.  Here’s our picture.  All graphs were created using Desmos Graphing Calculator:  www.desmos.com.  Desmos is a fabulous program!!

Getting Technical:  This graph is called an inverse variation (k = xy). K is the constant – which in this case is 13.1 miles.  I can’t get away from it.  If I’m going to complete a half marathon, I have to run 13.1 miles.

Why is it called an inverse variation graph?  If I decrease my rate/speed and walk very slowly (1 mile per hour) then my time increases.  But if I increase my speed and run super fast (9 miles per hour) then my time decreases.  As one element gets bigger, the other gets smaller.  And vice versa.

Exploring the Possibilities! 

I’m going to highlight 4 runners:  Joanie Benoit Samuelson, my former co-teaching partner, Jennifer Boundreau, Marge Simpson and finally, Dash from the Incredibles.

Joanie Benoit Samuelson: World class long distance runner.  1984 Olympic Gold Medalist – LA, 1984 marks the year, the women’s marathon became an official event at the summer olympics.  Two-time Boston Marathon winner, Chicago Marathon winner.  She current lives in Freeport, Maine. You can check out her website at:  www.joanbenoitsamuelson.com and Facebook page at:  www.facebook.com/JoanBenoitSamuelson2

Jennifer Boudreau: My former colleague and co-teacher, now a high school math teacher  and founder of the blog: runningwiththegirls.com   She started running seriously 8 years ago.  Since then, she’s completed 3 marathons, 7 half marathons, 14 10ks and 29 5ks!!!

Marge Simpson:  TV icon. I picked Marge because, I feel that she has the determination to complete a half marathon, but figured in that famous green dress of hers, she would only walk.

Dash:  His name says it all.  He’s fast.                                       Incredibly fast.

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Breaking Down the Math

Disclaimer:  Throughout the calculating process, I did round.  Therefore, the rounding has created slight discrepancies.

Marge.  If she walks at a rate of 3 miles per hour, it will take her over 4 hours to finish – Approximately 4 hours and 22 minutes.

Joanie Benoit Samuelson:  She’s in her late 50’s and is still a strong runner.  She clocked her last marathon sub 2 hours and 50 minutes.  Using that rate, her half time would be approximately 1 hour and 25 minutes.  Giving her a rate of 9.2 miles per hour.

In her prime, Joanie’s fastest marathon time was 2:21:21 when she won the Chicago Marathon in 1985.  Her speed was just over 11 miles per hour

.              

Jennifer Boudreau:  Jen’s fastest half time is 1:44:04 or 1.73 hours.  This means her rate is roughly 7.6 miles per hour.

Dash:  His speed inspired some conversation in my home.  After looking it up on the internet, I’ve decided to go with a speed of 766 mph.  He’d finish the half in approximately 1 minute.

The Graph

1. The curved line represents the constant (in this case a distance) of 13.1 miles.
2. The ordered pair represents (rate of speed, final time)
3. Technically, when the rate and the time are multiplied, the product should be 13.1
4. Since I rounded the information, the product will be close but not exact.                                                            Rate  x  Time = Distance 
   1. Marge (3, 4.367)                       3      x   4.367 =   13.101 miles
   2. Jennifer (7.6, 1.73)                   7.6   x   1.73   =    13.2012 miles
   3. Current Joanie  (9.2, 1.42)     9.2   x   1.42   =    13.064  miles
   4. Joanie (Prime)  (11, 1.19)       11      x    1.19   =    13.09  miles
   5. Dash (766, 0.017)                   766    x   0.017 = 13.022 miles
5. Where’s Dash?
   1. Let the students discuss this!
6. Students can create the graph themselves at www.desmos.com

Here’s Dash

Since Dash is so incredibly fast, I had to zoom out in order to see his ordered pair on the graph.  Before zooming out, have the students discuss where Dash would be located on the graph.  

Conclusion:

A lot of math topics are touched upon in this post.

1. Inverse variation (k = xy)
2. Distance = rate x time formula
3. Calculating a runner’s rate based on their final half marathon times.  (Jennifer, Joanie)
4. Calculating a runner’s time based on a given rate (Marge, Dash)
5. Substitution
6. Solving equations
7. Estimating the time using a table format
8. Converting hours in decimal form to hours, minutes and seconds
9. Graphing
10. Checking to see if the product of x and y values within the ordered pairs equaled the constant of 13.1 miles
11. Analysis of the graph (Where’s Dash?)

Initially, I simply wanted to show how a graph communicates, mathematically, the outcomes of a race.  As the post progressed, the initial idea developed into more and I can see how this post outlines a potential classroom project. Therefore I’ve listed, below, possible scenarios that students could explore.  Once all the graphs are completed, students can then compare how the constant effects the look of each graph.

My final thought

The next time you see:

   or     or     or   

Picture:

Other posts in this series:

Race Day (Part 2):  The Morning of…

Race Day (Part 3):  The Results:  Involving Inverse and Direct Variation Graphs

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Scenarios to explore with students:

NASCAR:  Daytona 500:  500 miles = rt

Commute:  25 miles = rt     6 miles = rt

Swimming:  100 meters = rt    200 meters = rt

Running:  26.2 miles = rt              5k = rt             400 meters = rt

Cycling:     56 miles = rt

Skiing:  Giant Slalom (Women 250 – 400m range  Men 250 – 450 m range)                                             300 meters = rt

Skating:  3000 meters = rt       1500 meters = rt

Walking:  4 miles = rt