Monthly Archives: December 2013

Names of Numbers

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

Screen Shot 2013-12-19 at 11.41.26 AM

. . . 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!

Screen time

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:

Screen Shot 2013-12-06 at 3.01.10 PM

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?

‘I’m not a math person’ no longer a valid excuse

This article was sent to me via the Jerry P. Becker listserv, and was originally published in the Business Insider on Monday, November 18.

Enjoy!

I’m Not A Math Person’ Is No Longer A Valid Excuse
By Kelly Dickerson
Contrary to popular opinion, a natural ability in math will only get you so far in studies of the subject.

Research published in Child Development found that hard work and good study habits were the most important factor in improving math ability over time.

But bad attitudes about math are holding us back.
Most of us would never think that “I’m bad at reading,” is a good excuse to stop taking English classes, so why is it ok, even normal, to say “I’m bad at math”?
A survey in 2010 conducted by Change the Equation found that three out of 10 Americans said they consider themselves bad at math. Over half of the 18 to 34-year-old bracket find themselves regularly saying they can’t do math. Almost one-third of Americans reported they would rather clean a bathroom than solve a math problem. [See http://changetheequation.org/press/new-survey-americans-say-%E2%80%9Cwe%E2%80%99re-not-good-math%E2%80%9D ]
And this math anxiety is a real problem: A study published in PLoS One in 2012 found that anticipation of doing math can actually affect the same regions of the brain that pain does. Essentially, math is painful. [See http://www.plosone.org/article/info:doi/10.1371/journal.pone.0048076 ]
Our attitude about math matters more than we think
Generally, people believe their learning ability works in one of two ways, according to research conducted by Patricia Linehan from Purdue University. [Seehttp://docs.lib.purdue.edu/dissertations/AAI9921102/  ]We classify our learning abilities in a given subject as “incremental orientation” – the belief that we can continually improve our ability by studying and practicing, or we think about our learning as an “entity orientation” – the belief that we can’t get any better no matter how hard we try. One person can have different orientations for different subjects.
Entity orientation toward math – basically saying, “I’m not good at math and so I never will be” – is a dangerous thing. When someone with entity orientation about learning math gets a math problem wrong, they think it’s just an indication of the poor math ability they were “born with,” according to a study published in Personality and Individual Differences in 2010. [See http://www.sciencedirect.com/science/article/pii/S0191886909003833 ]
This can have a very negative impact on motivation. If we don’t believe we can improve, we won’t bother trying.
Research shows that hard work, not natural ability, is the most important factor
The study mapped the progress of math ability in 3,520 students for five years – from grade five until grade 10. Students’ math ability was measured by their performance on the PALMA Mathematics Achievement Test. Questions included basic arithmetic, algebra, and geometry. The researchers also asked the students to answer questions about their study habits and interest in math.
In the early grades, a high IQ generally meant a high math score. But it turns out natural talent will only get you so far. How students study made a big impact on how much their math ability improved. Students who simply relied on memorization when studying, and didn’t attempt to make deeper connections with other areas of math, didn’t show much improvement over time.
The researchers also found that where a student’s motivation came from made a difference in their improvement. Students who said they wanted to get better at math simply because they were interested in the subject ended up improving more than those who wanted to get better in the interest of good grades.
“While intelligence as assessed by IQ tests is important in the early stages of developing mathematical competence, motivation and study skills play a more important role in students’ subsequent growth,” Kou Murayama, the lead researcher on the study, said in a press release [Seehttp://www.sciencedaily.com/releases/2012/12/121220080443.htm ]
You can see the difference it made in the chart to the left. Students listed as high-growth believed they could get better at math the more they practiced and used in-depth study techniques. Students listed as low-growth were more likely to believe that math ability is something you’re born with and it can’t be improved, and they relied more on memorization when studying.
How can we change our attitude about math?
Not only do we hear “I’m bad at math” from our peers, but we’re bombarded with messages that it’s OK to be bad at math. For instance, there are shirts made for young girls that check off shopping, music, and dancing as their best subjects, but deliberately leave the box next to math unchecked. There are also shirts that say “Allergic to Algebra” and “4 out of 3 people are bad at math.” [See http://www.sltrib.com/sltrib/money/56698622-79/math-shirt-girls-stores.html.csp ]
There are math-specific learning disabilities like dyscalculia – sort of the math equivalent of dyslexia – but this kind of learning disability does not explain poor performance in math in the general population. [See http://www.ncld.org/types-learning-disabilities/dyscalculia/what-is-dyscalculia ]
Psychologist Jonathan Wai said in a Psychology Today article that until we stop thinking being bad at math is funny, it will continue to be socially acceptable. [Seehttp://www.psychologytoday.com/blog/finding-the-next-einstein/201203/why-is-it-socially-acceptable-be-bad-math ]
Focusing on math as a skill, just like any other skill learned in school, could help increase our math literacy and encourage more young women and men to enter the field.
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See also Stereotypes About Math Are Holding Us Back  —   http://www.businessinsider.com/stereotypes-about-math-hold-americans-back-2013-11