# National Math Storytelling Day!

Did you know that today, September 25 is Math Storytelling Day!  Yahoo recommends celebrating this quirky holiday with the following Abbot and Costello video clip:

In the clip, the donut baker (Costello) is convinced that 7 goes in to 28, 13 times.  In fact, in trying to convince Abbott he’s correct, he’s able to show this three different ways!

In each explanation, Costello misuses the idea of place-value to convince Abbott that 28 divided by 7 is 13.  But guess what?  From the mathematics he does, what he’s really showing Abbott is that 28 divided by 7 equals 1 + 3, which is exactly what we expect (and know) 28 divided by 7 equals!

Let me show you:

Costello’s first attempt at 7 into 28 is this, 2 can’t be divided by 7 (he says).  But, remember the 2 doesn’t represent 2 ones; it actually represents 2 tens (more commonly known as 20) so if he wanted to he actually could divide part of 20 .  .  . but we’ll get to that later.  Right now, its just important that you remember that the 2 actually represents 20.

Then, he divides 7 into 8, which is 1 with a remainder of 1 (he does this correctly).  So, he asks his sous chef to give him back the 2 (which is really 20) and writes “21.”  Now he proceeds to take 21 divided by 7, which is 3 and tells Abbott that 7 into 28 is 13, BUT 1 and 3 are both in the “ones” place meaning that 7 into 28 actually equals 1 + 3, or 4.

Costello really does this:

Now, I have three questions for you:

1. Can you give similar explanations for the other two methods shown in the video?

2. If you were Abbott, how would you have convinced Costello he was incorrect?

3. Remember if 28 divided by 7 did equal 13, that means that we could take 28 things and divide them into 7 groups each containing 13 (in this case) donuts.  Suppose Costello really did take his 28 donuts and make 7 groups, each with 13 equal-sized donut pieces.  What fraction of a donut would each person get?

# And the Winner is . . .

So, did you see that last week some lucky person in South Carolina won the \$400 million Powerball Jackpot? (Technically, it was \$399.8 million, but \$400 million is close enough!)

It seems like every time there’s a big jackpot won in the lottery you read statements like “you’re more likely to be struck by lightening,” or “you’re more likely to marry a prince,” or in the case of the CNN story about this particular Powerball winner “you’re more likely to get struck by lightening and bitten by a shark.” (Talk about a bad day!)  They also go an to say that the chances of winning a Powerball Jackpot are 1 in 175223510.  Don’t you wonder how people come up with all of these statistics?

Let’s take a look at how Powerball is played . . .

According to the Powerball website, lottery numbers are drawn from two drums.  The first drum contains 59 white balls and the second drum contains 35 red balls (These red balls are all potential Powerballs).  The jackpot is won by matching all five white balls in any order and the red Powerball.

In order to calculate the odds of winning, we need to figure out the odds of matching all five white balls, in any order, and the red ball.

I like to think about situations like the one described above by picturing an empty (in this case) lottery ticket.  Like this:

The first ball that pops up could be any of the 59 balls, the second ball could be any of the 58 balls, the remaining blank spots on the ticket will be filled by drawing from the final 57, 56, and 55 balls respectively.

That looks like this:

Now, the order doesn’t matter in the way I arrange the balls, remember?  That means if the winning white balls are 1, 2, 3, 4, 5 and my ticket is 2, 3, 4, 1, 5; I’m on my way to winning the Powerball!  So now we need to figure out how many different ways the 5 white ball numbers can be arranged.

The first white number could be any of the 5 numbers drawn from the drum, so I have 5 choices for the number in the first spot.  I only have 4 remaining numbers for the second spot, 3 for the 3rd spot, etc . . .

Because I can rearrange the numbers and still have a winning ticket, the possible number combinations I need to win has just been decreased!  Now, we can calculate the number of ways to get a winning combination from the white balls in the Powerball drawing:

The total number of combinations is :

And now for the red Powerball!  The process we used above is going to be the same for red Powerball, except instead of having to match 5 numbers you only have to match 1 and because there’s only one number to be matched it doesn’t really make sense to talk about whether or not the order matters . . . there’s only one number.

Since you know a method to use and you know the answer, I’m pretty confident you are going to be able to figure how CNN could report that a person has a 1 in 175223510 chance in winning.

Good Luck!

P.S. I used a few different techniques from Discrete Mathematics or Counting Theory in this post that I didn’t explicitly name.  First, as in the case of the white lotto balls I was choosing 5 balls from a collection of 59 balls.  The order in which I arranged these 5 balls was not important.  This situation describes a combination or a binomial coefficient.  There is a formula associated with these types of situations.  We used this formula, although I did lots of canceling to make the numbers used look less overwhelming.  You could rewrite the situation that I described above as a combination using the formula provided from the combination link and see if you can get it to look like the one I used.

Also, we calculated the factorial of 5, denoted 5!.  If you attempt the challenge I’ve given you above, you’ll want to make sure you know what a factorial is.

# Staying Motivated in Online AP Courses

The Iowa Online Advanced Placement Academy (IOAPA) allows Iowa students to take APTM classes online.  IOAPA is especially meant for rural schools that do not have the resources to support APTM classes.  Educators can learn more at www.belinblank.org/ioapa

There is no getting around it—online APTM courses can be difficult to complete.  This can be especially true for students whose drive dwindles during the semester.  With that in mind, how can IOAPA students boost their motivation?

Create a realistic schedule

Given that IOAPA students must structure study time more so than students who take classroom-based courses, it is crucial that they create realistic work schedules.  A mistake commonly made by teenagers and adults alike is to assume tasks take a shorter time to complete than they actually do.  Students should thus plan sufficient time for each course assignment.  Furthermore, they must recognize when to say no to activities…

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# Fibonacci Number Sequence and Prime Numbers

Please don’t let this title turn you away from the post.  Its a really great post, I promise . . . I just couldn’t come up with something cute and catchy today!

Anyway, I’ve been very vocal about the fact that NCTM asks great questions on their Facebook and Twitter pages, but today I have another Tweet that I really enjoyed from Maths Jam:

So, this is a great question, right?  You probably know what makes a number a perfect square (just in case you don’t, look here), but you might not know about the Fibonacci Number Sequence.  I sort of wish that we’d talked about it before today, because there are many great and interesting things about this sequence of numbers, but we’ll just have to talk in more detail about those great and interesting things later.  For the purposes of this question you just need to know that the Fibonacci number sequence is a sequence of numbers which is generated by adding the previous two numbers together.  So terms 1 – 5 of the Fibonacci Number sequence are:

1, 1, 2, 3, 5, . . .

It seems that this particular Tweet asserts that the number sequence is actually

0, 1, 1, 2, 3, 5, . . . which I’d never seen before, until I did a little investigating via Wolfram Mathworld, but it isn’t really central to the answering of this question.

So, just to recap; we’re looking for numbers in this sequence that are also perfect squares.  In that case, I guess we should start by listing the Fibonacci Numbers, starting with the “0” term:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, . . .

Here’s the thing about answering this question, the number sequence generates an infinite number of terms meaning that, yes–I would venture a guess that at some point there will be another perfect square number in this sequence of numbers, but I’m not sure what it is.

This person calculated and factored the first 300 Fibonacci Numbers; and from this list it looks like 0, 1, and 144 are the only Fibonacci Numbers that are also perfect squares.

There are proofs, involving Lucas Numbers that also show that the only Fibonacci Numbers that are also perfect squares are 0,1,144.  That means that with the use of mathematics we are able to prove things about numbers that we know exist, but that no one has discovered yet.  Wow.  Just wow.

# Mark-Recapture

Last month my husband and I decided to take our kids to a local state park for a picnic and some fishing.  We had scouted the best places in the park to fish and took off for an afternoon adventure.

For those of you who don’t have a lot of experience fishing with a 7, 5, and 1-year old time is of the essence.  You have a very important 15 minute window in which to have someone catch their first fish.  If you don’t get any action in about the first 15 minutes you can pretty much bet on the fact the the kiddos you brought fishing will be ready to put away the fishing poles and start to build a fire for the s’mores.

Well on that fateful day in the middle of August, that 15 minute window slipped right through our fingers.  And then, the 15 minutes turned into 30 without so much as a nibble.  My husband was able to entertain the kids a little while longer by letting each of them practice casting their own finishing line (while everyone else crouched in the bushes to avoid an accidental snag), but even the novelty of doing that wore off about an hour into the fishing expedition.

Soon, the excitement of going on a fishing trip turned in to:

“When can we make s’mores?” –My 5 year old

“How come that person is catching fish?!?”–My 7 year old

“Dad are you sure there are even fish in this lake?”–My 5 year old again

(Just as a side note we stayed at the park for 3 hours and did not catch ONE. SINGLE. FISH., but to make up for it we made s’mores and then stopped for ice cream on the way home.)

Anyway, this fishing trip reminded me of a method for estimating the fish population in rivers and streams called the Mark-Recapture method.

The Iowa DNR has a really nice explanation of the Mark-Recapture method, although the whole using real crickets thing creeps me out a little; but this is basically how it works:

1. Catch fish and tag them (or Mark them).  Record the number of fish you’ve caught and tagged.

2. Go fishing again.  Record the total number of fish you catch and the number of caught fish that you tagged previously (that’s the Recapture park).  Then set up the following proportion to estimate the population of fish in the the lake/river/stream you were fishing.

Now that you know the basics of Mark-Recapture, lets think about a few things:

1. Taking a census of the fish in the lake is not practical in this case.  Why?

2. If I’d really like this to be an accurate picture of the population of fish in this lake, what are some things that I will have to consider when collecting the data (ie fishing, tagging, and counting)?  Think about things like location of the fisherman, time of day for fishing, weather conditions during the Mark-Recapture data collection.  What are other items that should be considered?

3. Why might the Iowa DNR or any naturalist for that matter be concerned with the fish population of a particular lake or stream?  How can the Mark-Recapture method help address those concerns?

P.S. If you’re an AP Statistics Teacher or Student, this is a great sampling example to consider.  Did you know sampling questions occur the most frequently on AP free response exams, and they are also the most often missed on the exam too?  Don’t believe me?  Check out this article.

# I’m an IOAPA Mentor: Where Can I Find Resources and Support?

The Iowa Online Advanced Placement Academy (IOAPA) allows Iowa students to take APTM classes online.  IOAPA is especially meant for rural schools that do not have the resources to support APTM classes.  Educators can learn more at www.belinblank.org/ioapa

Mentors for APTM courses will inevitably run into bumps as they assist students in learning material and in completing coursework.  Fortunately, a range of resources are available to help mentors navigate each semester.  We offer a few suggestions below.

Participate in Professional Development

Mentors involved with IOAPA are strongly encouraged to attend the Belin-Blank Center’s Advanced Placement Teacher Training Institute (APTTI).  This week-long summer workshop prepares teachers to develop and teach APTM courses.  Mentors may also peruse a comprehensive directory of institutes and workshops designed for APTM teachers and mentors.  For instance, refresher courses will be held in Des Moines in late October (see directory…

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