Jimmy Fallon’s Twelve Days of Christmas Sweaters tradition has become something I look forward to each December. The sweaters are one-of-a-kind masterpieces. I love when the sweaters are revealed. Jimmy reaches into a bright red Christmas stocking and randomly pulls out a number, the seat number of the winner of the sweater. Miraculously, the winner of each sweater looks fabulous in it, no matter how big or small the winner is. I love this tradition, the sweaters, the winners modeling the sweaters, but I also love hearing the seat numbers. Each seat number has something special about it. (Just because it is a number!) By day three, I knew I wanted to blog about the numbers this year. The seat numbers were 295, 257, 314, 270, 419, 126, 256, 417, 433, 242, 232, and 120. I immediately knew something special about several of the numbers, but some of them I had to research. Can you figure out what is so special about each one?
Three of the seat numbers were primes. Which three?
One of those primes is both the fourth Fermat prime and the second-largest known Fermat prime. Which prime number is that?
Two of the numbers were palindromes (numbers that read the same forward and backward). Which two?
One of the seat numbers is equal to 1 × 2 × 3 × 4 × 5. Mathematicians write that as 5! Which seat number is equal to 5!?
One of the numbers is 10π rounded. Which one?
How Do Some of the Seat Numbers Shape Up?
Two of the numbers were decagonal numbers. Which two?
126 is not only a decagonal number, but it is also a pyramid formed by stacking the first six pentagonal numbers on top of each other.
1 + 5 + 12 + 22 + 35 + 51 = 126.
120 comes in THREE shapes.
One of the seat numbers is a star:
Something Special About Each Seat Number:
I’ll explain some of these reasons below.
Three of the Numbers Were the First Numbers to do Something Special:
242 is the smallest number whose square root can be simplified that is followed by three other numbers whose square root can also be simplified. Also, all four numbers have exactly six factors. Numbers with exactly six factors always have simplifiable square roots.
242 = 2·11²; its six factors are 1, 2, 11, 22, 121, 242.
243 = 3⁵; its six factors are 1, 3, 9, 27, 81, 243.
244 = 2²·61; its six factors are 1, 2, 4, 61, 122, 244.
245 = 7²·5; its six factors are 1, 5, 7, 35, 49, 245.
417 is the smallest number that is the first of four consecutive integers that are divisible by a different number of primes.
419 is one less than 420, the smallest number divisible by 1, 2, 3, 4, 5, 6, and 7. As a consequence of that, 419 is the smallest number that leaves a remainder of 1, when it is divided by 2, a remainder of 2, when it is divided by 3, a remainder of 3, when it is divided by 4, a remainder of 4, when it is divided by 5, a remainder of 5, when it is divided by 6, and a remainder of 6, when it is divided by 7. This next graphic is a different way to make the same point.
What Do I Mean by Sum-Difference?
Two of the seat numbers have factor pairs that make sum-difference: the numbers in one of its factor pairs add up to a particular number and the numbers in a different factor pair subtract to the same number. Coincidentally, both of the seat numbers are related to 30, another number that makes sum-difference.
I’ve so enjoyed discovering what made all those seat numbers special, and I hope that you have enjoyed reading about them as well!
Since this is my 1705th post, I’ll write a little about that number, as well.
Factors of 1705:
1705 is a composite number.
Prime factorization: 1705 = 5 × 11 × 31.
1705 has no exponents greater than 1 in its prime factorization, so √1705 cannot be simplified.
The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1705 has exactly 8 factors.
The factors of 1705 are outlined with their factor pair partners in the graphic below.
More About the Number 1705:
1705 is the hypotenuse of a Pythagorean triple:
1023 1364 1705, which is (3-4-5) times 341.
Plinko is a fun carnival game of chance. This Plinko board is really just a portion of Pascal’s triangle. OEIS.org informs us that 120 is the smallest number to appear six times in that triangle. Why did those six times happen?
120 = 10!/3!/7! That’s why it appears twice in the 10th row of the Plinko board below.
120 = 16!/2!/14! Which is why it appears twice in the 16th row as well.
120 will appear two more times in its 120th row.
Now step right up and learn some other incredible facts like
120 = 5! because 1·2·3·4·5 = 120
120 is also the smallest positive multiple of 6 that is neither preceded nor followed by a prime number!
What kind of shape is 120 in?
120 is the 15th triangular number because 15(16)/2 = 120,
it’s the 8th tetrahedral number because (8)(9)(10)/6 = 120 (That means 120 is the sum of the first eight triangular numbers), and
it is the 8th hexagonal number because (8)(2·8-1) = 120.
Math Journals and Creative Writing
Every Playful Math Carnival contains blog links about ways to play with math and insights into teaching math. Blogging about math helps clarify thoughts, document experiences, and share the joy math brings us. It is a lot like keeping a math journal. Denise Gaskins wrote a post about the benefits of math journaling and included some prompts to help students get writing. Whether writing about joys or frustrations, math journalling has its benefits.
Abhishek Pathania wrote a clever limerick titled Maths that uses mathematical terms such as chance, calculated guess, multiply and divide. I enjoyed the limerick and I bet your students will, too. Another blogger, Roland, shared Maths Limerick, which is quite a bit of fun, too.
The next stop at our carnival is a house of horrors that is simply terrifying to some people. It is known as . . . . .
In Life Cameo’s post Learning, a young girl goes from liking math to feeling significantly less confident and quietly suffering from math anxiety. Thankfully her teacher intervened and she is now just starting to understand it again.
A young man named Dave blogged about his lifelong struggles with math in Dealing with Learning Disabilities in Math. Although he occasionally used a strong word to voice his frustrations, his is an important point of view that ought to be shared. This school year I am working with students who need specialized help with mathematics so this post gave me some food for thought.
Which method is better for children to learn math, discovery or traditional? The Intrepid Mathematician suggests a combination of the two and how to implement that teaching in A third path for early math education.
Mathematical Art on Exhibit
The average preschooler/kindergartner only gets 58 seconds of math instruction a day. Those who get Paula Krieg to teach them for one fascinating hour a week are really fortunate! You can see what I mean by reading her post, Little Hands, Little Books, Folds, & Math.
Number Loving Beagle shares a raw, personal story of years of yearning for artistic talent in Math is Beautiful (and other lies). Math really can make beautiful, frameable art as demonstrated in that post, but too often math has become nothing more than misery-inducing, anxiety-producing, seemingly worthless calculations. Which math will you choose for yourself and your children?
Su Leslie created a beautiful piece of fractal art in Pretty Maths. Su’s work could inspire others to see the beauty in mathematics.
Rachel Shey shares some more mathematical art and thoughts in the post Math and Art. I also liked her thoughts about two fields intersecting.
The Math Museum featuring Calculators, Castles, and Puzzles
Simona Prilogan of Let’s Math regularly posts a number puzzle on her blog, Let’s Math. Some of the puzzles may be easier to solve than others, but I’m sure students will be able to figure these two out. Boats Tuesday Maths Puzzle and Sunshine Thursday Maths Puzzle. That second one actually contains a few carnival pictures!
I visited a type of museum inside Romania’s Corvin Castle in Hunedoara this summer. Although I didn’t know when I visited, Hunedoara is Simona Prilogan’s hometown! I was delighted to find a post she published about the castle in her poetry blog less than a month before I wrote a post with some mathematical pictures from inside the castle. I am amazed at how small the world of mathematics can be!
Life Through a Mathematicians Eyes also grew up in Romania and offers a guided tour of Calculators That Made History. When I took the tour I was amazed at how old some of those calculators are. I’m sure you will enjoy the tour very much!
Colleen Young has several different mathematical examples in her post Here’s the diagram. What’s the question? What better way could there be to learn any of those topics frontward and backward than make it feel like solving a puzzle?
BloggingIsAResponsibility wrote a post titled Is Math Meaningless, and Is That an Insult? If you’re introducing syllogisms in your geometry class, you might want to try some of these effective but meaningless arguments!
Life Through a Mathematician Eyes offers thoughts and study videos on more advanced Logic Problems beginning with Studying Logic – Day 1.
Musings of a Mathematical Mom blogged about a mathematical adventure her children enjoyed. They counted and divided using Christopher Danielson’s book How Many. Her children even drew pictures afterward that would allow them to count and think about even more fractions. Who could ask for anything more?
Life Through a Mathematicians Eyes reviews three books that teens and teachers can most certainly enjoy in New Book Discoveries. The books reviewed are Weird Maths: At the Edge of Infinity and Beyond by David Darling and Agnijo Banerjee, Your Daily Maths:366 Number Puzzles and Problems to Keep you Sharp by Laura Laing, and 50 Maths Ideas You Really Need to Know By Tony Crilly.
Susan mentioned Ramanujan and the book The Man Who Knew Infinity when she wrote a blog post she called The Story of the Locked Box and the Key of Dreams. Her title sounds like a mathematical fairy tale, but it is not a storybook at all. It gives a vivid description of her lucid mathematical dreams, her struggles with dyscalculia, and her triumphs in learning math. Ramanujan also had wonderful mathematical dreams, so she is in good company.
Dealing with histograms might seem as treacherous as getting through an obstacle on American Ninja Warrior, but Math Only Math gives step by step histogram instructions to help middle and high school students navigate through those different-height rectangles in record time.
Finally, no matter where or how you teach mathematics, remember these words Jennie penned in An Open Letter to Teachers, “You have to share your love and passions. That’s your joy. In that way, you are sharing you. And, all that children want to know is that you love them and love what you are teaching. If they know that, the floodgates will open to learning.”
The future of mathematics education is in YOUR hands. Have fun!
This was my 1200th post. Here are some facts about the number 1200:
1200 is a composite number.
Prime factorization: 1200 = 2 × 2 × 2 × 2 × 3 × 5 × 5, which can be written 1200 = 2⁴ × 3 × 5²
The exponents in the prime factorization are 4, 1 and 2. Adding one to each and multiplying we get (4 + 1)(1 + 1)(2 + 1) = 5 × 2 × 3 = 30. Therefore 1200 has exactly 30 factors.
Taking the factor pair with the largest square number factor, we get √1200 = (√400)(√3) = 20√3 ≈ 34.64102
1200 is the hypotenuse of two Pythagorean triples:
336-1152-1200 which is (7-24-25) times 48
720-960-1200 which is (3-4-5) times 240
1200 is the sum of twin primes 599 and 601
1200 looks interesting to me when it is written in some other bases:
It’s 3333 in BASE 7 because 3(7³ + 7² + 7¹ + 7⁰) = 1200,
550 in BASE 15, because 5(15² + 15) = 1200
363 in BASE 19, because 3(19²) + 6(19) + 3(1) = 1200
300 in BASE 20 because 3(20²) = 1200, and
220 in BASE 24 because 2(24² + 24) = 1200