Tag Archives: Patterns

Thunderstorms!

It’s safe to say that thunderstorm season has officially arrived in Iowa!  The temperature and the humidity has been on a steady climb for the last couple of weeks (remember when we were making jokes about how cold it was?!?) and seasoned midwesterners can spot the ideal weather for a good thunderstorm from miles away!

I love thunderstorms!  For some reason, they always prompt me to bake a batch of chocolate chips cookies whenever they roll through!  (There’s nothing quite like watching the clouds roll in while you chow down on homemade cookie dough!)  Unfortunately, my children do not share my affinity for thunderstorms, not even the promise of warm chocolate chip cookies can calm their nerves when the thunder starts booming and the lightening flashes!

Last week a quick thunderstorm rolled up in the middle of dinner.  Instead of focusing on the scary booms and flashes I said to them “Did you know if you count the number of seconds between when you see the lightening and hear the thunder, you can estimate the distance the thunderstorm is from our house?”  (P.S. Did you know that?)

The speed of sound through the air is approximately 340 meters per second, and the speed of light is approximately 300 million meters per second.  Even though thunder claps and lightening  flashes are happening at the same time, the difference in speed makes it seem as though the lightening is flashing before the thunder.

Using the relationship between distance, rate, and time we know that D = R*t, where D is distance, R is rate, and t is time.  Since we have the rate of sound and light in meters and seconds, we’ll also report D and t in terms of meters and seconds.

Now, suppose you hear thunder approximately 5 seconds after you see a flash of lightening.  If we use the relationship between distance, rate, and time we can substitute known values into the equation, which gives us D = 340*5 = 1700 (remember this is meters).  1700 meters is approximately 1 mile.

The next time a thunderstorm rolls up in your neighborhood, see if you can track how quickly its  moving through the area.  Keep a record of the length of time between lightening flashes and thunder rumbles.  Can you tell when the storm is getting closer and farther away from you?

P.S. I got my facts and figures from two great sources: the National Weather Service and The Department of Physics at the University of Illinois Urbana Champaign.

 

 

Boston Marathon Times

This morning thousands and thousands of people did something I can not even imagine doing . . . they ran the Boston Marathon (It was actually 35,671 entrants to be exact)!

This year the winning men’s time was 2:08:37 (Meb Keflezighi from California) . . . that’s an average speed of about 1 mile every 4.88 minutes.  (As a comparison I re-started Couch-to-5K last night . . . and I ran about 1 mile every 11 minutes).

Anyway, the whole Boston Marathon thing got me thinking . . . I wonder how Meb’s time compares to other people who have won the Boston Marathon?

The first Boston Marathon was run in 1917.  John J. McDermott (NY) won that race with a time of 2:55:10.  He was still averaging about 1 mile every almost 7 minutes.  So, is Meb just exceptionally fast?  Was John just exceptionally slow?

4.21.14

The graph above represents all of the Boston Marathon times–from John to Meb and all of the marathoners in between.  What do the data seem to tell you?  Was John exceptionally slow?  What about Meb?

Averagetime

This shows the average time for 10 year time spans of Boston Marathon winners.  What seems to be happening to marathon times-over time?

If you had to model Boston Marathon winning times, based on the number of years since the first marathon what type of model would you use?  Exponential Growth/Decay?  Linear Increase/Decrease?  Quadratic model?  Why?  Do you think there might be anything noteworthy about the graph as people continue running the marathon?  Will anyone ever run the marathon in under 2 hours?  1 hour? (if someone ran a marathon in under an hour they would be averaging 1 mile approximately every 2.25 minutes)

I’d love to know what you think!  In the meantime . . . I’ll be trying to get under the 10 minute mile mark with my Couch-to-5k app!

52 Week Reverse Savings Plan

Yesterday I came across this picture taken by one of my former high school students:

photo

I didn’t really know what the 52 week reverse savings plan was, but based on her hashtags and the amount of money she deposited yesterday it seemed reasonable that the goal was to save money each week (there are, after all 52 weeks in a year) and that she would decrease the amount of money she was depositing into her savings account by one dollar each week.  (Turns out I was right).

I can’t be certain, but I think the idea behind this type of savings plan is that you capitalize on the idea that at the beginning of the year, right after you’ve made your New Year’s resolution, you’re more likely to set aside larger amounts of money for the program and as the program continues, you can talk yourself into saving the next week, because its less then you set aside the week before.  I’d venture a guess that the reverse of this 52 week reverse savings plan would not be as effective.

Her photo made me think of a story I used to tell my Pre-Calculus and Algebra II students when we began talking about series of numbers.  The story is this (and I think its loosely based on a true story.  Read it here):

Carl Friedrich Gauss is a famous mathematician, and as is true with other young geniuses, his elementary school teachers found young Carl to be quite annoying and unruly.  Why?  You might ask.  Well, for two reasons really.  First, Carl could finish the work intended to take 30 minutes in 5, thus spending the remaining 25 minutes doing what young children do when they’re bored.  Second, Carl seemed to be able to outsmart his teachers in almost everything.  One day at school the same scenario that had been playing out for days once again played out in young Carl’s classroom–his teacher had given an assignment and Carl had finished in a fraction of the time the assignment was meant to take.  As he began to distract and disrupt his other classmates, his teacher had a brilliant idea!  She called Carl up to her desk and told him to add all of the integers from 1 to 100.

Imagine his teacher’s surprise (and probably frustration!) when Carl came back to her desk a mere minute later with the correct answer!

When questioned about what he had done he laid out the following pattern for the teacher:

Screen Shot 2014-01-07 at 10.50.55 AM

So,

Screen Shot 2014-01-07 at 10.51.51 AM

But now I’ve added the numbers from 1-100 twice, so to account for this I really need to write:

Screen Shot 2014-01-07 at 10.52.33 AM

Isn’t that clever?

And, can you tell how this relates to the Instagram pic posted by one of my former students?  It seems to me that it would be reasonable to ask how much money she will have saved by the end of 2014.  One way we could answer this question would be to add money deposited each week:

52+51+50…+3+2+1

But, that seems a little tedious and thanks to Carl Gauss we can do this more efficiently.  Namely:

Screen Shot 2014-01-07 at 10.55.55 AM

(Similar to my M&M posts (here, here, and here), I smell a series (ha!-get it, series?) of posts related to this topic. For example, how many weeks does it take to save half of the money from the 52 week challenge?  If my student is depositing this money into a savings account, then she’s earning interest.  If she leaves the money in the account until she goes to college in two years, how much money will she have?  Is the amount really all that different if she only saves for half the year?  Or every other week? . . . the possibilities are limitless (ha! ha!-get it, limitless?)) (Check out the second post in the series here.)

NCTM Circle Problem

If you’ve been here before, you probably know I really like the NCTM problems they post on their Twitter and Facebook pages.  Well they posted one last week that I just couldn’t seem to stop thinking about (doesn’t that usually mean you’ve got a good question on your hands!?!)

In case you missed it, here was the question:

NCTM Circle Picture

What’s the value of the question mark?

The first time I saw the picture I immediately noticed that 4 and 8 were on the circle and that they were only one wedge apart.  I really, really wanted the question mark to be worth 16, but then that would mean that the 13 should have been a 32 and that the 23 should have been a 64 and that the 92 should have been a 128 . . . clearly I was wrong.  (P.S. can you tell what I was going in order to fill in the wedges of the circle?).

Then, I noticed that there were a few prime numbers on the circle (remember how to check to see if a number is prime or composite (via Math Warriors)?).

Here’s my tip . . . anytime you’re looking for a pattern in a situation prime numbers should make you a little suspicious.  You should be suspicious because they don’t mix well with other numbers, when considering “typical” pattern generators such as multiplication, division, or power properties.

Given my prime number predicament I felt like I had two choices . . . I could look for a pattern generated by adding or subtracting the numbers or I could look at the prime factors of the numbers in the circle and see what happened.  I wasn’t really compelled to try the whole add/subtract thing because from looking at the entries, that didn’t seem very likely.

Instead rewrote the numbers on the circle as a product of their prime factors and I came up with something that looked like this:

photo

Based on my little diagram, I was able to fill in the missing value.

If you happened to come across this question via Twitter or Facebook last week, you know there were two common answers floating around.  The first common answer was 120.  The second was 79.

1. How did people arrive at both of these answers?

2. Do you think that one is “right” and one is “wrong”?  If so, why?  Take a side and convince me the other answer is incorrect.

Another point that was brought up online was that if you started at different points on the circle, you could get a different question mark value.

3. What do you think about that argument?