Category Archives: Number Theory

Happy Pi Day!

How could a blog devoted to all things math not have a post on Pi Day?  Truthfully, I had big plans for today . . . starting with the Pi Day cookies I wanted to bake and bring to work:

This is as far as I got making Pi cookies for today.

This is as far as I got making Pi cookies for today.

#pidayfail

Honestly though, I’ve been running around like a crazy woman these last few weeks and just didn’t have enough time to get my self together to have a meaningful Pi Post today!  It seems as though the day as been jinxed.  You saw how well my Pi cookies turned out, then I went to the store to buy a pie to bring to work and they didn’t have any!  My husband did come through for me though and snagged this awesome Pi shirt from his junior high school (Thanks awesome junior high teacher!).

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Since I don’t have a great Pi Day post for you today, I thought I’d round up some of my favorite Pi activities from around the web . . . former students of mine will know some of these activities well 🙂

A great Pi Day cartoon:

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Pi set to music.  ( know there are lots of these around the web, but I have found this one to be the best!)

The argument for Tau Day (with two pies . . . I’m in!)

My favorite activity to do with students on Pi Day . . . although I’ve heavily adapted it!

Happy Pi Day one and all!  I’m off to round up some pie for lunch 🙂

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Happy Valentine’s Day 143

1 4 5 11

Do you?

It just makes me think of Valentine’s Day.  As a little kid I remember getting them from my parents, in their little cardboard box.  In high school, I used to tell my sweetheart of the month that all I wanted for Valentine’s Day was one of those little boxes . . . forget the flowers and stuffed bears!

What’s that you say?  You don’t know what 1 4 5 11 means?  It’s code.  1 4 5 11 is code for I love Necco Sweethearts.  You know those chalky little hearts with the Valentine sayings on them?  I didn’t make up this code, Necco did.  Have you seen these hearts?

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(Find the top 14 conversation hearts here, including 143).

Do you know what 143 stands for?  I (1) love (4) you (3).  Get?  If not don’t feel bad.  This person Facebooked Necco to find out why in the world she ate a heart with 143 on it!

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Anyway, I think its cute and all but as far as codes go . . . its really not that great.  I mean 143 could stand for lots of things couldn’t it?

But here’s the thing, Necco’s touching on something that mathematicians have used for a long time.  That is, numbers as code for something else.  Try this one:

91215225251521.

OK, OK.  So its not that hard, right?

Its either:

IABAEBBEBEAEBA or ILOVEYOU.  I’m not sure that I would want to send some sort of top secret code through cyberspace if my coding technique was A=1, B=2, C=3, etc.  But what about this code?

4560776126257605

Do you want a hint?  OK.  It looks like that number might be divisible by 5.  Oh!  It is divisible by 5.  I wonder what you get when you divide the number by 5?

Hmm.  We might be on to something.  It seems to me that a great way to code something might be to do the whole A=1, B=2, C=3 thing and then to multiply it by another number.  If the person I’m sending the code to knows the number to divide my code by, the code is pretty darn easy for the receiver to crack and its fairly difficult for a spy to intercept and figure out what it says, don’t you think?

So, find a sweetheart and send them something in code for tomorrow.  My sweetheart is getting this message.  Can you crack it?

22125135191513519235520851182019

Math Dice Game

My mom bought me this little game called “Math Dice” for Christmas this year.  Have you heard of it?  I hadn’t, and truth be told, I’m not sure my mom realized it was a game she was buying it for me.

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When I opened the package she said “I thought you could do something creative with those dice and your math blog.”  In the days after Christmas, I scooped the unopened box of dice into our “junk drawer” (sorry mom) and rediscovered them this weekend while cleaning.  On the back the of the box were the directions to the “Math Dice Game.”

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This morning I had a little extra time, so I thought I’d give the game try!

Step One: Roll the 12-sided target dice

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Step Two: Roll the three 6-sided scoring dice.  Combine the three scoring dice in anyway to match or come closest to the Target Number.

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Ummm, really?  1, 1, 2?  The closest I could get to 80 was 24.  This is what I did:

(1+1+2)! = 4*3*2*1.  Can you get closer?

My second roll of the Target Dice was 36

My Scoring Dice roll was 6, 4, 2

Super easy: (6*(4+2))=36.  Did you get 36 another way?

My last roll was 20

And my scoring dice were 6, 3, 1

I couldn’t get 20.  I could get 18 and 21, but 20 right on the money was a little tricky.  Can you do it?

Much, much more to come about this fun Math Game, with my new Math Dice . . . I’m working on a table of possible Target Number combinations as we speak!

52 Week Reverse Savings Plan

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

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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:

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So,

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

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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:

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(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.)

Names of Numbers

This morning on Twitter, This Day in Math posted this:

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. . . it made me wonder about two things:

1. How many other palindromic primes are there between 1 and 365?

2. What in the world in a palindromic prime?

(I know these questions probably seem out of order, but what can I say . . . that’s the order I thought of them!)

Even though I thought of the questions in backwards order, clearly we need to figure out what a palindromic prime is before we can figure out the number of palindromic primes between 1 and 365.

Do you know what a palindrome is?  Its a word that spells the same thing forward and backward.  For example my sister’s name, Anna is a palindrome!  And, if you’re a reader of this blog you certainly know what a prime number is (if you want to read more about the primes on this blog check here and here).

Given the definitions of palindrome and prime, one could reasonably assume that a palindromic prime is a prime number whose digits are the same when written forwards or backwards (and one would be correct).

Here’s the question I was left with though . . . Is a single digit number a palindrome?  (And as long as we’re on the topic, is a single letter word a palindrome?–Believe it or not, someone wrote an article about this).  According to Google, the answer is that single digit numbers or single letter words are palindromes, but people don’t usually talk about them because they aren’t interesting.

For the purpose of counting the number of palindromic primes between 1 and 365, however, I am going to consider single digit prime numbers to be palindromic primes (if this offends you, just cross these numbers off of your list . . . there are only 4 of them for goodness sake!)

Anyway, palindromic primes between 1 and 365.  It seems to me that first it would make sense to list all of the prime numbers between 1 and 365.  (We’ve talked about slick ways to test for prime-ness here).  So, using the methods we already know list away!

Then, look for the palindromes in your list of primes.

(I’ve starting a short list for you.  Here’s what I have 2, 3, 5, 7, 11–This means Jan. 2, Jan. 3, Jan. 5, Jan. 7, Jan. 11 are all palindromic prime-numbered days in any given calendar year)

How many did you find and what are the corresponding dates?

You know, this Twitter post, led me to think about another question.  My favorite type of named numbers are perfect numbers.  A perfect number is a number that equals the sum of all of its factors (not including the number itself).  For example 6 is a perfect number, because the factors of 6 are 1, 2, 3, and 6 and 1+2+3 = 6.

So, how many perfect numbers are there between 1 and 365?  And what are the corresponding dates in a year?

As an side note . . . my son’s due date was 06/28/06 and I was really excited!  Can you guess why?

Happy Counting!

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It’s been a while since I’ve written about a mathematics problem from NCTM and I kinda of miss writing about them (You can read the other ones here and here)!  Last week they posted a question that only got snarky answers on Facebook, so I decided this would be the perfect one to write about 🙂

Here’s the post:

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From the snarky answers it seems to have drawn criticism because of the number of movies Ryan supposedly has, but I just have to say from my own experience that my children could certainly given Ryan a run for his money in the movie collection department!

Anyway, Ryan’s movie collection is apparently quite extensive.  Also, he only likes movies that are fairly short (1 hour and 30 minutes in total).  If Ryan were to watch these movies back to back to back to back to (you get it, all 70 without stopping); how many days would it take him to watch said movies?

Can you tell right off the bat that this is a conversion situation?

My game plan:

Convert hours and minutes to the same unit of measure; then use this unit of measure to calculate the number of days of the marathon movie watching!

Except that as I was typing this, I came up with another game plan . . . each pair of movies equals 3 hours.  Then, I don’t have to worry about converting minutes to hours or hours to minutes.  AND, 70 is an even number of movies, meaning I can make pairs of movies without leaving any movies out.  AND, 3 hours is a really nice value to have when dealing with days, because there are 24 hours in a day and 3 divides 24 evenly.  So, I can watch 8 pairs of movies a day (or 16 movies a day).

New Game Plan:

16 goes in to 70 little more than 4 times (actually it goes into 70, 4.375 times)

That means Ryan will need to watch 8 pairs of movies for 4 days.  At the end of 4 days he will have watched 64 movies, leaving him the final 6 movies (or 3 pairs of movies, or 9 hours of movies) for the 5th day.

So my final answer is 4 days and 9 hours of movies to watch all 70 movies!

Just to double check, you could do the converting I described at the beginning of the post . . .OR you could describe a different method to do the calculations if the mood strikes you!  If you do this another way, I’d love to hear about it in the comments below!

I also feel the need to address the posts on Facebook about Ryan’s movie situation.  Most of the comments were along the lines of “only a mathematics website, textbook, teacher, etc., etc. would ask such a silly question.  This question isn’t realistic.”  I must say I have to respectfully disagree.  The first thing I thought about when I read this question was the Brita water filter commercial showing plastic water bottles stretched across the Earth’s surface.


If this commercial had been written as a word problem in a mathematics class, it probably would have sounded a little something like this:

The makers of Brita water filters claim that 1 filter is able to filter the equivalent of 300, 16.9 oz. bottles of water.  Suppose Ryan uses 1 Brita water filter per month for one year.  If Ryan had used plastic bottles, instead of water filters how many times could the number of plastic bottles used wrap around the equator of the Earth?

I’m willing to admit that this type of problem situation isn’t one that we encounter in our every day lives; however these types of conversion situations come up whether they be for impact, or marketing, or something else.  Isn’t nice to know that you . . . reader of the It’s Just Math Blog have a way to evaluate such statements for their accuracy?

And finally, with regard to Ryan and his movie watching I’m willing to bet that the following problem statement may have been a little more well-received.  Why?  You might ask?  Because most of us know someone who’s done this 🙂  If this problem statement had been written about me it would have been about watching Law and Order SVU marathons 🙂

Suppose Ryan is a huge fan of James Bond movies.  In fact, this weekend he plans to watch every James Bond movie ever made back-to-back.  Will he have enough time to watch all of the movies if he starts Saturday morning at midnight and finishes Monday morning in time for school at 8am?