Monthly Archives: August 2013

30 Day Blogging Challenge

OK!  I’m doing it!  I’ve read about lots and lots of blogging challenges over the past few days and I’ve been trying to decide whether or not I want to try one.  And . . . I’m doing it!

(Thanks to The Nester and her blog post about her October 31 Day Challenge)

I’m going to be a mentor for the Student Blogging Challenge.  Just for fun, I’m also going to post each day for the month of September.  I’ll keep all things Student Blogging Challenge related on the 30 Day Blogging Challenge page, but I’ll reference it in some of my other blog posts also.

I hope you’ll join me!  What a fun way to start the school year!

Restaurant Tipping

About a week ago I came across an article online explaining that some restaurants were doing away with having customers tip the waitstaff.  Instead, this particular restaurant increased the cost of each meal on the menu by 15% and asked patrons not to leave tips on the table (in fact, when customers still left tips they were chased down by waitstaff to return the money!).

As you can imagine, the article interviewed many restauranteurs, a few on either side of the tipping debate.  One particular restaurant owner, who was in favor of the no-tip movement explained his support this way:

Screen Shot 2013-08-19 at 1.50.05 PM

Calculus?  Really?  To figure out how to leave a tip?  Now, we know he didn’t really mean calculus.  We know he was just trying to say that for some people, figuring out how much of a tip to leave can be complicated.  In fact, its so complicated that people who are out trying to enjoy a nice leisurely meal shouldn’t have to be bothered by “doing the math” so to speak.

But here’s the thing, doing restaurant math isn’t hard.  

Let’s say you go out to eat with your friends and the bill is $50.00.  If you’re leaving tip that’s 10% of the total, you leave $5.00

So, the 10% is going to be the baseline that we work off of.  (You know how I got $5.00, right? . .  . Move the decimal 1 place to the left).  Now, we know that $5.00 is 10% of the total; so if we want to leave a 20% tip we should double the amount and leave $10.00.  If we want to leave a 5% tip, we should leave half of that amount (in this case $2.50).  How, what if we wanted to leave a 15% tip?  I know 10% is $5.00 and 5% is $2.50, so 15% must be $7.50.

If you look at this “tipping table” posted on CNN money (written by Emily Post), you’ll see that once you can figure 5%, 10%, 15%, and 20% of your total bill (always excluding tax) you can figure out a tip in any given situation.  No calculus (or cell phone calculator) needed!

Screen Shot 2013-08-19 at 2.15.34 PM

Now, you might say:
“Yeah, but you picked a nice round number for your example. My restaurant bills are never like that!”

OK, OK.  Good point!  So let’s say your bill was $23.18.  Now, take 10% of the total ($2.31).  The 20% tip is easy ($4.62).  The 5% might be a little tricky, only because $2.31 doesn’t divide evenly . . . that’s OK.  Call it $2.30, half is $1.15.  So 15% would be about . . . $3.45 or you could just be nice and round it to $3.50 (or you could very, very precise and leave a tip of $3.46).  But see, you can do it!

Now, go out dinner and practice your new-found tipping skills!

Iowa Butter Cow

I’ve lived in Iowa for 27 of my 32 years, but until 2 days ago I had never been to the Iowa State Fair.  It’s true what they say, you can get everything on a stick and the fair is larger then anyone could imagine (well, until you’ve been there of course!)

photo

(My children watching some sort of cow show.  You can tell the 2-year was enthralled; the other   2 were humoring their little sister)

The Iowa State Fair frequently shows up on National and Families with Kids Summer Bucket Lists.  In fact, this year Al Roker visited the fair!

One of the main attractions at the fair is the Butter Cow.  Its a sculpture of a cow made entirely of Iowa Sweet Cream Butter (with a little help from wood and wire forms underneath it all!).

I’d heard about the Butter Cow before.  I’m sure I’d even seen a picture or two of the cow, but as soon as we got through the gates and on to the fair grounds I told my family the first thing we must see is the Butter Cow!  So my family of 7 (my mom and dad were with us, also visiting the Iowa State Fair for the first time) walked right past the bacon wrapped ribs on the stick, and the hand-dipped corn dogs (OK, OK so we made an impromtu stop at the mini cinnamon roll stand . . . you would understand, if you could have smelled them) to see the famous Butter Cow.

We could tell where the cow was, before we even saw it because of the steady stream of people walking by the large window to catch a glimpse of the cow.  But, when we finally got there this is what we saw:

IMG_0216

Pretty impressive, if you think about it.  The butter cow is about 5 feet 6 inches tall and 8 feet long.  It weighs approximately 600 pounds (a really dairy cow weighs about 1000 pounds)  According to the Iowa State Fair website the butter cow could butter about 19,200 slices of toast and would take the average person 2 lifetimes to eat.  These statistics got me thinking:

1. If this cow could butter 19,200 pieces of toast, how much butter is being put on each piece of toast?

2. If it would take the average person 2 lifetimes to eat the butter cow, how much butter per day is the “average person” eating?  And, what makes a person “average”?

3. In the video about the 2013 butter cow artist Sarah Pratt says that this year’s cow weighs between 450 and 50 pounds.  If one stick of butter weighs 4 ounces, approximately how many sticks of butter were used for this year’s butter cow?

Want to know more about the Butter Cow?  Check out this YouTube video produced by Iowa Public Television.

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1912 Math for 8th Graders

A version of this post first appeared at kkdegner@blog.com on 8/12/13.

This morning when I got to work, I noticed that “1912 eighth grade exam” was trending on Yahoo!:

Screen Shot 2013-08-12 at 2.31.10 PM

I was intrigued.  So I began look and came across this article in the Washington Post.

They show a recently recovered test for 8th grade graduation from 1912.  (You can see a picture of the entire exam in the Washington Post, I’m showing you the arithmetic part of the exam below).

Screen Shot 2013-08-13 at 9.55.19 AM

It looks like there were a variety of subjects 8th grade students were supposed to be familiar with, but I was really interested in question 8 in the arithmetic section.

It says:

How long a rope is required to reach from the top of a building 40 ft high to the ground, 30 ft from the base of the building?

I was interested in this question because I just happen to have in these in my office:

picstitchbooks

That’d be my great grandma’s Arithmetic, Geometry, and Algebra textbooks from when she was in junior high and high school. And, that’d be a picture of her work doing polynomial division, as well as her perfect attendance for the month of December 1917.  The copyright on the books range from 1879 – 1911, so the material in the books was written right around the time that this eighth grade graduation exam was given.

Can we also just recognize how much of a mathematical rockstar my great grandma must have been!?! (My understanding is that she was also a great oatmeal chocolate chip cookie and Chex Mix maker . . . all of the important things in life; math and food)

Anyway, I was interested in looking through her textbooks, because I was interested in trying to figure out how students in 1912 would have been expected to solve this problem.  First, if we can agree that the situation described creates a right triangle, then I think we can consider that this problem could have been answered one of three ways; by recognizing and memorizing the Pythagorean Triples, using the Pythagorean Theorem, and by using Trigonometry.

Now, if you have the Pythagorean Triples memorized, you know that the smallest right triangle (in terms of perimeter), with all side lengths being integers is a 3-4-5 right triangle.  Then, the triangle formed in this situation is a 30-40-50 right triangle (the smallest P. Triple triangle, times 10), so the rope must be at least 50 ft long.

But, her textbooks didn’t make any mention of the P. Triples (I was a little surprised, I thought math was all about looking things up in tables back then).  But, her Plane Geometry textbook did have this:

photo copy 2

and this:

photo copy

leading me to believe that these test-writers probably expected students to use the Pythagorean Theorem, also giving a minimum rope length of 50 ft.

The last way to address this problem would be by using trigonometry, but I doubt this was the case.  I didn’t find trigonometry in any of her textbooks.  I know people knew about trigonometry in 1912, but my guess is that branch of mathematics was reserved for the students in upper level mathematics classes, and that, that particular method of solving probably wasn’t expected on the 8th grade exit exam in 1912 (or now, for that matter).

Also, trying to answer this question using trigonometry certainly isn’t incorrect, but it makes this question a multi-step one.  However, just in case you’re interested here’s the way we could have used trigonometry to help answer our question:

tan ß=30/40; ß≈0.6435; sin(0.6435)=40/x; then x=50.

Not that you could find them, but if you’re interested here are the citations for the textbooks I consulted:

Hart and Feldman (1911). Plane Geometry. American Book Company. NY, NY.

Milne. (1911). First Year Algebra. American Book Company. NY, NY.

Ray. (1856). The Principles of Arithmetic, Analyzed and Practically Applied for Advanced Students.  Van Antwerp, Bragg & Co. Cincinnati.

Wells. (1909). Essentials of Algebra for Secondary Schools. D.C. & Co Publishers. Boston.

 

Map Coloring

This post first appeared at kkdegner@blog.com on 8/5/13.

This week I needed a map of all of the counties in Iowa for a little work I was doing in my office.  A quick Google search turned up quite a few options:

iowa-county-map

One of the first things I noticed about the maps was that the counties that were touching each other were each colored a different color.  This makes sense really, if all of the counties were the same color, there really wouldn’t be any point in coloring them at all, now would there?  Similarly if every other county was colored a different color (that is in the East/West direction), it might make it hard to differentiate between the counties that touch in the North/South direction.  So that leads me to the question; Why four colors for the map of Iowa’s counties?  Could it have been done with three colors?

The other thing I noticed about the maps was that if the counties shared a side, the colors were different, but if they shared a vertex (i.e. if they were diagonal from each other) this wasn’t necessarily the case.

Look in the top left corner of the map, for example.  I see a green to green vertex, a blue to blue vertex, yellow to yellow vertex, as well as an orange to orange vertex.  Being the purist that I am, I don’t really appreciate the fact that these colors are touching each other.  Which leads me to my next question . . . can I color the map of all of the counties in the state of Iowa only using four colors, such that all sides and all vertices don’t touch another county colored in the same way?

So, I set off to answer my questions with state of Iowa coloring pages and colored pencils:

picstitch

First, to tackle the problem of coloring the map using only three colors.

Twice, I got off to a promising start.  I used a technique that graph theorists call the Greedy Algorithm (Harris, et al, 2000).  That is, I used as few colors as possible until the third color was needed.  In other words, I was greedy with my colored pencils; not wanting to use more than I needed to when coloring the graph.  My use of the Greedy Algorithm and my coloring first horizontally and then vertically resulted in the following map colorings:

photomap     photomap3

In both cases, I ran in to trouble when one of the counties was bordered by more than four counties, see?

Then, I decided to try to tackle the second question which was “Can I color the map of all of the counties in the state of Iowa using only four colors, such that all sides and all vertices don’t touch another county colored in the same way?”  After a false start:

photomap2

I answered my question.  And the answer was yes!  (No, I’m not showing you my map.  Print off an Iowa map and start coloring.  You’ll get it, I promise!)

The colorings of the maps I’ve described are all proper colorings (Crommwell, 2000).  The Four Color Theorem tells us that only four colors are needed to properly color any map, but a natural question might be:

Are four colors always needed?  Can some maps be properly colored using only three colors?  Or two colors?

What do you think?  I think you’d better break out some colored pencils and get to work!

Playing with Plato

This post first appeared at kkdegner@blog.com on 7/29/13.

This weekend I came across this article, in the Iowa City Press Citizen (Sunday, July 28, 2013 edition to be exact).

Which made me feel a little nostalgic, because when I used to teach high school and middle school I used to teach my students to do this:

photo[1]

And when I saw the article this weekend I was reminded about how interesting these shapes can be, so interesting in fact it made me find and read these:

photo

And then, it inspired me to write a blog post (lucky you!).

In the article the author, Kim Cook, says that geometric shapes are all the rage for fall decorating.  The shapes can be used as lamp shades, and knick-nacks, and even wall paper!  Cook lists numerous places to find these geometric shapes, and even gives a suggestion about how to make your own.  (This DIY blogger also tells you how to make your own 3-D geometric shapes).

Well, would you believe me if I told you that this isn’t the first time 3-D geometric shapes are all the rage in decorating?  In fact, the use of regular polyhedra, is routinely found in the work of Renaissance artists and craftsman.

leonardo-dodec      pacioli

Taken from http://www.georgehart.com/virtual-polyhedra/

And, the aesthetic  appeal of these shapes is probably one of the main reasons the Greeks became obsessed with studying them.  Its also probably the same reason so many people in the year 2013 are willing to put them on their mantels (even if they don’t really know anything about them!).

So, let’s talk a little more about these 3-D geometric shapes that may soon be coming to an “en vogue” home near you.  The most basic of these are the regular polyhedra.  (Sometimes called the Platonic Solids).  The solids are made by having the same number of the same regular polygons meet at each vertex (the regular polygons are called faces).  For example, the regular polyhedron with the least number of faces is the pyramid.  This solid is made up of four regular (or equilateral) triangles, three meeting at each side.  Following this same pattern, we get another regular polyhedra, the octahedron; which is made up of 8 regular triangles, meeting at each side.  You keep playing the game, until regular triangles don’t make a solid anymore (why would that happen?).  Then, we can start with regular quadrilaterals (commonly known as squares).  How many square faces can meet at each vertex to form a solid? Next, we can check regular pentagons and so on . . .

Go ahead and try this and while you’re trying this think about these few things:

1. Why must there be a minimum of 3 faces meeting at each vertex?

2. How do you know when you can’t make anymore solids out of the given shape?

3. Regular polyhedra can only be made out of a few regular polygons, why?

We could talk for days about this topic, but that’d make for a pretty long blog post, so let’s agree that this is a good starting point and revisit the idea of regular polyhedra (and polyhedra in general) at a later date, ok?

The books in my photo at the top of this post include:

Combinatorics and Graph Theory by John M. Harris, Jeffry L. Hirst, and Michael J. Mossinghoff. Springer. 2000.

The Crest of the Peacock by George Gheverghese Joseph. Princeton University Press. 2000.

The Joy of Mathematics by Theoni Pappas. Wide World Publishing. 2004.

Mathematics and its History (2nd Edition) by John Stillwell. Springer. 2000.

Polyhedra by Peter R. Cromwell. Cambridge University Press. 1997.

Finding Prime Numbers

This post first appeared at kkdegner@blog.com on 7/22/13.

Last week I came across this little doozy on my Facebook page, which was posted by The Belin Blank Center, which it looks like they must have reposted from the National Council of Teachers of Mathematics

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I love the question for so many reasons!  First, I love it because its not complicated.  If you know what a prime number is and you know how to find square roots, you’re set!  Trial and error might be the weapon of choice for many people answering this question.  And while there’s nothing particularly wrong with trial and error, it can take a while and, well, its not really that interesting.  But, we could use just a little, tiny bit of strategy to try to cut down on the amount of numbers we have to test in order to answer the question.

To address this question we have to check 3 things 1)is the number prime? 2) is the square root less than 20? 3)is there a larger prime number whose square root is also less than 20?  Let’s start by setting an upper bound to eliminate the 2nd and 3rd questions.  We know that 20 squared is 400, meaning all numbers less than 400 have a square root less than 20.  So now we know we’re looking for prime numbers which are less than 400.  If you’re feeling very adventurous, you could just list all of the numbers between 1 and 400.  The largest prime in that list of numbers would be the answer to the question.  But, do you really want to check 399 numbers too see if they’re prime? No?  Me neither.  (We don’t have to check 1, we know its not prime or composite).

Good news!  We can use our knowledge of numbers and their factors to whittle this list down pretty quickly!  First, get rid of all the even numbers (except for 2–its prime), because we know they’re divisible by 2.  Additionally, get rid of all of the numbers ending in 5 (except for 5–prime!), because they’re divisible by 5.  Now we’re down to, what 161 numbers by my count?  That still seems like a lot of numbers to test, no?  There are still other numbers we could systematically check off of our list of possibilities, for example the multiples of 11 and 3 are pretty easy to spot (prime and prime, respectively).  Taking the grand total of possibilities down to 105 numbers.

photo[1] copy

From here, we could do a few things.  Starting with the largest number left on the list (the largest number I have left is 397), we could check to see if the number is prime.  If it is, then great!  We’ve found the answer. If you don’t like that method we could continue along the path we’ve been on.  That is, now cross out all of the multiples of 7, 13, 17, and 19.  See what’s left standing, check to make sure that its prime and just for fun, calculate an approximation of the square root.  Happy Number Crunching!

Wait, you didn’t think I’d actually give you the answer, did you?  What would be the fun in that?  I’ll tell you what I will do though, I’ll tell you about a little tool/algorithm/rule that is very helpful in answering this question (and its one I’ve been using in this post).  Its called the Sieve of Eratosthenes and it says this:

List all of the numbers from 1 – n (in our case 400).  To find all of the prime numbers, start with the smallest prime (2) and cross out every second number in your list.  Now, go to the next number in the list that you haven’t crossed out (3).  Cross out every 3rd number (notice, some of the numbers like 6 and 12 in your list you will have now crossed out twice).  Continue this process through all of the numbers 1 – square root of (n).  The numbers remaining are all of the prime numbers 1 – n.  

Now you can answer that nice little problem!

What’s that you say?  You don’t understand why 1’s not a prime number?  Don’t worry, you wouldn’t be the first to ask.  Have you ever heard of Math Warriors?  Felicia does a great job of explaining this question in this episode.  (If you haven’t seen Math Warriors yet, you might as well just start at the beginning by going here.)

P.S. I included lots of great math resources in this post.  Follow them on Facebook and Twitter!

BBC Facebook                              BBC Twitter @belinblank

Math Warriors Facebook         Math Warriors Twitter @MathWarriors

NCTM Facebook                         NCTM @NCTM

Sources:

Weisstein, Eric W. “Sieve of Eratosthenes.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/SieveofEratosthenes.html