131 Playful Math Carnival

Welcome to the Playful Math Education Blog Carnival featuring the amazing prime number 131, whose digits can mutate into other prime numbers right before your eyes!

131, a permutable prime number

make science GIFs like this at MakeaGif
Yessiree, 131 is prime, and so is 113 and 331. Do I need to mention that 3, 11, 13, and 31 are also prime numbers?
131’s next trick happens when you add up all the 2-digit PRIMES that begin with a 4:
41 + 43 + 47 = 131.
Because 131is a palindrome, it reads the same forwards and backward. Here’s another trick: 131 is 65 in BASE 21 and 56 in BASE 25.

 

We have many different attractions this month. You can go to any category quickly here:

Carnival Attractions:

Arithmetic

You’ve heard of the three R’s, reading, and writing and ‘rithmetic, but what is arithmetic? Joseph Nebus shares a few comics about basic arithmetic and explains what they mean:

Arithmetic is also television’s Lisa Simpson’s favorite subject in school and she will miss it greatly as she recovers from the mumps. In this blog post, Safi explains Dr. Hibbert’s comforting words to her about polygons, hypotenuses, and Euclidean algorithms.

Art

You can always count on Robert Loves Pi to produce a beautiful and complex geometric design. This one he calls Two Rhombic Polyhedra with Tessellated Faces. Here’s another one:

Paula Beardell Krieg helped students create big, beautiful geometric artwork and origami in Summer Projects with Teens.

Also, check out Paula’s Paper, Books, and Math Workshop for many more ways to learn math through art.

Big Prize, Little Chance of Winning

Several years ago Mental Floss wrote about carnival games that offer big prizes but have very little chance of being won. This carnival has a couple of those as well. They are called unsolved math problems. Even if winning probably isn’t going to happen, that doesn’t mean the games and activities aren’t fun. Explaining Science updates us on a very famous unsolved problem, The Goldbach’s Conjecture. Supercomputers have worked on it, but we are no closer to a solution.

In A Neat Unsolved Problem in Number Theory That Kids Can Explore, Mike’s Math Page explores the new-to-me Collatz conjecture that for every positive n, the sum 3 + 8n will equal a perfect square plus an even number. It’s a simple enough conjecture for kids to understand and it is fascinating, yet mathematicians have not been able to prove or disprove it yet!

Creative Writing

Subha laxmi Moharana (Angel Subu) writes creatively about some tough topics in high school mathematics in Math Poem. I think her words could be turned into a rap.

Poetrywithmathematics shares Doug Norton’s lovely mathematical poem Take a Chance on Me.

What if graphs were self-conscious about their looks? High School aged students can consider that thought as they read the imaginative blog post, To Infinity and Beyond.

Displays

There’s a cozy classroom place that promotes mathematics in Our New Math Space. It was designed for older students by Continuous Everywhere But Differentiable Nowhere and includes many pictures.

Have you considered displaying a weekly math joke? MathEqualsLove shares a fun joke and a puzzle for kids to gather around and enjoy.

Factoring Quadratics

Super Safi uses another episode from the Simpsons to teach about the quadratic formula.

Food for Thought

Anybody can cook or do math. Really? What does that even mean? Math4Love explains both in What We Mean When We Say, “Anyone Can Do Math.”
Math with Bad Drawings makes a similar point in The Adventures of Captain Math.

Games

Joyful Parenting made a simple kindergarten-age counting game and called it Snack Math, but even older kids might enjoy figuring out exactly how many crackers are required to play the game.

How many are in the jar. What is a good estimate? Add Steve Wyborney’s clues one by one to get an even better estimate. He has 51 New Esti-Mysteries that also happen to teach several different math concepts.


For older students, Kent Haines a free game he calls Last Factor Loses. I played it a few times with a student. Making prime factorization a game really did make it more fun.

Geometry

Bn11nb enjoys the geometry of architecture. The pictures in this post are worth a look and could be an inspiration to your students.

House of Mirrors (Reflecting on Mathematics Teaching)

We often reflect on the effectiveness of our teaching methods. Sometimes we are advised to require students to use more strategies. We might ask them to notice or wonder about a concept. These two thoughtful posts will certainly give you cause for reflection:

“The More Strategies, the Better?

Noticing and Wondering: A powerful tool for assessment

 

Robert Kaplinsky shares ten things he’s embarrassed to tell you. Has he been reading your mind and mine?

Money

What is your favorite part of a cupcake? What if you could buy just that part? What if you wanted to put a whole cupcake together? How much would that cost? Your child can learn about money and decimals exploring those answers with Mathgeekmama’s  Money Math Problems.

Museum of Mathematics

Beads can be a fun manipulative when learning mathematics. Joseph Nebus has begun his 2019 Mathematics A-Z series by writing about the Japanese abacus. He compares it to a slide rule and the Chinese abacus. He also describes how to use it to add, subtract, and multiply numbers. Students could have some fun using it to understand place value, too.

Life Through a Mathematician’s Eyes is giving museum tours in A History of Mathematics-August. K-12 students could be fascinated by the mathematical relics from the Smithsonian founded in August 1846 as well as the Seven Bridges of Königsberg solved by Euler in August 1735.

Pumpkin Patch

Erin of Sixth Bloom’s Pumpkin Math-Preschool Activity will engage your little ones as they learn to count and sort pumpkin-shaped macaroni or candies.

They will also love decomposing numbers using pumpkin seeds and  Mathgeekmama’s cute Pumpkin cards.

Posters

Digital Educators Alliance offers free posters of admirable women in math and related fields:

While Sara Van Derwerf set of 112 New Math Fail Posters will delight students as they notice and wonder about and LEARN from grown-ups’ computing mistakes.

Puzzles

7Puzzle gives some clues about a 3-digit number. Can you figure out what it is?

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Alan Parr writes about a newspaper puzzle called Evens Puzzles. He suggests that students can make their own and hints that he has thought up several variations of it. I look forward to reading about those!

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American Calendars for September had more than a week’s worth of palindromes. Would palindromes make a good puzzle? Yes! Print off a 100 chart and try Denise Gaskins’s A Puzzle for Palindromes. Also, check out her new Morning Coffee feature each week for more math teaching tips.

Next Month’s Carnival

That’s it for this month’s Math Education Blog Carnival. The 132nd Carnival will be next month at Arithmophobia No More. Would you like to share a post or host the carnival? Go to Let’s Play Math for details!

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1409’s Super Power

 

Stetson.edu informed me that 1409⁸ is the ONLY known 8th power that is the sum of EIGHT 8th powers. Wow! That seems to me to give 1409 quite the superpower!

What were those eight 8th powers that are included in the sum? That’s a puzzle more suited for a computer than a human, but Wolfram Mathworld Diophantine came to my rescue with this POWERFUL fact: 1324⁸+1190⁸+1088⁸+748⁸+524⁸+478⁸+223⁸+90⁸=1409⁸.

Go ahead and check it out on your computer’s calculator. It’s true! Notice also that two of those eighth powers are permutations of each other!

I was so intrigued with 1409 that I had to make this cape so everyone can see how super 1409 is:

Sometimes 1409 wears a more modest super cape because 1409² is also the sum of TWO squares:
159² +1400² = 1409² 

Here are some more super facts about the number 1409:

  • 1409 is a prime number.
  • Prime factorization: 1409 is prime.
  • 1409 has no exponents greater than 1 in its prime factorization, so √1409 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1409 has exactly 2 factors.
  • The factors of 1409 are outlined with their factor pair partners in the graphic below.

How do we know that 1409 is a prime number? If 1409 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1409. Since 1409 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1409 is a prime number.

1409 is the sum of two squares:
28² + 25² = 1409

1409 is the hypotenuse of a primitive Pythagorean triple:
159-1400-1409 calculated from 28² – 25², 2(28)(25), 28² + 25²

Here’s another way we know that 1409 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 25² = 1409 with 28 and 25 having no common prime factors, 1409 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1409. Since 1409 is not divisible by 5, 13, 17, 29, or 37, we know that 1409 is a prime number.

 

1408 Powers of 2 in the Multiplication Table

number, puzzle, factors, factor pairs, prime factorization,

I have a 10 × 10 multiplication table poster in my classroom to help students who haven’t memorized the times’ table yet. We have to spend our time going over more advanced topics. One student struggled with the idea of raising two to a power. I went to the poster and boxed in all the powers of two on it. While I boxed them in, I recited, “2⁰ = 1, 2¹ = 2, 2² = 2×2 = 4, 2³ = 2×2×2 = 8, 2⁴ = 2×2×2×2= 16, 2⁵ = 2×2×2×2×2= 32, 2⁶ = 2×2×2×2×2×2=64.”

I liked the pattern those powers of two made on the poster so I made this 32×32 multiplication chart on my computer and continued the pattern.

I expect the chart has many things for you to notice and wonder about. You could also do it with powers of 3, or another number, but you would need to use a much bigger multiplication table to show as many powers.

Now I’ll tell you a little bit about the number 1408.

1408 is not a power of 2, but it is 11 times a power of 2, specifically, it is 11 × 2⁷.

  • 1408 is a composite number.
  • Prime factorization: 1408 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 11, which can be written 1408 = 2⁷ × 11
  • 1408 has at least one exponent greater than 1 in its prime factorization so √1408 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1408 = (√64)(√22) = 8√22
  • The exponents in the prime factorization are 7 and 1. Adding one to each exponent and multiplying we get (7 + 1)(1 + 1) = 8 × 2 = 16. Therefore 1408 has exactly 16 factors.
  • The factors of 1408 are outlined with their factor pair partners in the graphic below.

Here is a festive multilayered factor cake for 1408:

So delicious! And here is a nicely balanced factor tree showing all of its prime factors:

 

1407 Please Stop Making Excuses for My Dear Aunt Sally

Please Excuse My Dear Aunt Sally. You’ve heard math teachers say that phrase many times. Supposedly, Aunt Sally is supposed to help you remember Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction as the correct order to do operations when simplifying math problems.

I say, please stop making excuses for my dear Aunt Sally!

My Dear Aunt Sally. People think they know her, but too often they really don’t. Lots of people have tried to please her. Sometimes they succeed, but just as often they fail. She seems to relish the fact that so many people misunderstand her.

I clearly remember my first year teaching a classful of seventh graders at a new school. I was trying to develop a good relationship with my students and be the best teacher I could be. One of the first lessons I was supposed to teach them was order-of-operations.

I wish I knew about the better mnemonic PEMA back then, but I didn’t. Instead, I brought my dear Aunt Sally to class with me: I introduced her to my students and tried to make it clear that multiplication and division were equals so they must be done in order from left to right whichever one comes first. The same is true of addition and subtraction.

“That’s not what we learned last year!” students responded. Their teacher last year brought Aunt Sally to class, too. but she gave them the impression that all multiplication was supposed to be done before any division, and the same for addition and subtraction. Yeah, Aunt Sally went to class their previous year and didn’t say a word when their teacher gave them misinformation. Now that I was telling them the truth about her, she didn’t speak up and tell them I was right either. Instead, she allowed me to lose credibility with my students that day as I insisted on sticking with the truth. If I had retold the lie, the students would have believed me more. I also discovered that for some problems in the textbook, you would get it right either way.

I seriously couldn’t believe that their teacher from the last year would have given them the wrong information. Surely the students misunderstood what had been taught. However, since that day, I have heard more than one teacher incorrectly tell students to do all the multiplication, division, addition, and subtraction in that order from left to right. Those teachers put the students’ next teachers in a catch-22:

That is why I prefer to keep “my dear Aunt Sally” away from kids. She always shows up at the beginning of the school year when students and teachers are trying to start off on the right foot.  She torments students and immediately causes them to feel bad about themselves or mathematics. She makes them question the teaching of their current teacher or their past teachers. She gets a kick out of making children and even adults feel like there’s no way to understand math:

Why do we allow Aunt Sally to abuse children like this? I want to shout, “please, stop making excuses for my dear, Aunt Sally!”

Let me tell you the story of when I decided not to introduce this abusive aunt to children every again.  It was 2016. I was substituting in a 5th-grade class. I wrote an expression I saw on twitter on the board and told the students it was my favorite order-of-operations problem. Here’s what I wrote:

10 + 9 + 8 × 7 × 6 × 5 – 4 + 321 = 

I, along with my dear Aunt Sally,  encouraged the students to figure it out. The students knew that 8 × 7 was 56. I watched them struggle to multiply 56 by 6 and then by 5. When I mentioned that they could multiply the 6 and the 5 first to get 56 × 30 to make the problem easier, they argued that doing that wasn’t allowed. They said that the order-of-operations demanded that the multiplication be done in ORDER from left to right.

They thought that order-of-operation makes multiplication no longer commutative?!!  How do you counteract that misinformation? After that day, not only do I not invite my dear Aunt Sally to meet my students, but I also avoid the phrase “order-of-operations”!

Order-of-operations is just an ALGORITHM! It doesn’t trump the commutative property, and it doesn’t even have to be used to solve these kinds of problems!

Jo Boaler’s tweet especially applies to this kind of problem and this algorithm.

Besides, are these kinds of problems still necessary since typing on a computer no longer has the same limitations as typing on a typewriter? I hope you think about that! If you insist on using an algorithm, I suggest you use PEMA instead.

Since this is my 1407th post, I’d like to tell you a little bit about that number:

  • 1407 is a composite number.
  • Prime factorization: 1407 = 3 × 7 × 67
  • 1407 has no exponents greater than 1 in its prime factorization, so √1407 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 1407 has exactly 8 factors.
  • The factors of 1407 are outlined with their factor pair partners in the graphic below.

1407 looks interesting when it is written in some other bases:
It’s 111333 in BASE 4,
21112 in BASE 5, and
727 in BASE 14.

1406 Has a Very Cool 4th Root

To find the 4th root of 1406, all you need to do is take its square root twice. The square root of 1406 is 37.4966665185. . .

Take the square root of that and you get a decimal starting with 6.12345…

That’s pretty cool. I’m glad Stetson.edu let me know about it!

Here’s a little more about the number 1406:

  • 1406 is a composite number.
  • Prime factorization: 1406 = 2 × 19 × 37
  • 1406 has no exponents greater than 1 in its prime factorization, so √1406 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 1406 has exactly 8 factors.
  • The factors of 1406 are outlined with their factor pair partners in the graphic below.

1406 is the sum of the first 37 EVEN numbers because 37 × 38=1406.

1406 is the hypotenuse of a Pythagorean triple:
456-1330-1406 which is (12-35-37) times 38

1405 is the Sum of Squares

I knew that 1405 was the sum of two consecutive squares, but Stetson.edu let me know that it was the sum of even more consecutive squares, ELEVEN to be exact!

Because it is the sum of the 26th and the 27th squares, 1405 is also the 27th centered square number. Here are 1405 tiny squares illustrating that fact:

Here’s more about the number 1405:

  • 1405 is a composite number.
  • Prime factorization: 1405 = 5 × 281
  • 1405 has no exponents greater than 1 in its prime factorization, so √1405 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1405 has exactly 4 factors.
  • The factors of 1405 are outlined with their factor pair partners in the graphic below.

I’ve mentioned one of these before, but 1405 is the sum of TWO squares in TWO ways:
27² + 26² = 1405
37² + 6² = 1405

1405 is also the hypotenuse of FOUR Pythagorean triples:
53-1404-1405 calculated from 27² – 26², 2(27)(26), 27² + 26²
444-1333-1405 calculated from 2(37)(6), 37² – 6², 37² + 6²
800-1155-1405 which is 5 times (160-231-281)
843-1124-1405 which is (3-4-5) times 281

1404 Texas Tessellation

I recently visited family members in Texas. My daughter-in-law is awesome at both mathematics and quilting. My photo does not do her work justice, but Texas is tessellated in this quilt! She also carefully chose the fabrics she pieced together. Do they remind you of anything for which Texas is famous?

Someone else designed the pattern, but piecing these pieces together was not the easiest sewing project.

I wondered if anyone else had thought to tessellate Texas and found a couple of examples on twitter. As this first one asked, should we call this Texellation?

Now I’ll tell you something about the number 1404:

  • 1404 is a composite number.
  • Prime factorization: 1404 = 2 × 2 × 3 × 3 × 3 × 13, which can be written 1404 = 2² × 3³ × 13
  • 1404 has at least one exponent greater than 1 in its prime factorization so √1404 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1404 = (√36)(√39) = 6√39
  • The exponents in the prime factorization are 2, 3, and 1. Adding one to each exponent and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 × 4 × 2 = 24. Therefore 1404 has exactly 24 factors.
  • The factors of 1404 are outlined with their factor pair partners in the graphic below.

1404 is the hypotenuse of a Pythagorean triple:
540-1296-1404 which is (5-12-13) times 108

Since 1404 has so many factors, it also has MANY different factor trees. Here are four of them mixed in with some Texas tessellations!

Prime Factorization of the Hundred Numbers up to 1400

Almost one-third of the numbers from 1301 to 1400 have 4 factors. Only 1/5 of the numbers have 8 factors.

Since 1/3 is significantly bigger than 1/5, the amount of factors for these numbers wouldn’t make a very exciting horse race. Here is the breakdown:

  • 11 numbers had 2 factors
  • 1 number had 3 factors
  • 32 numbers had 4 factors
  • 7 numbers had 6 factors
  • 20 numbers had 8 factors
  • 2 numbers had 10 factors
  • 13 numbers had 12 factors
  • 4 numbers had 16 factors
  • 1 number had 18 factors
  • 2 numbers had 20 factors
  • 5 numbers had 24 factors
  • 1 number had 28 factors
  • 1 number had 32 factors

The rosy looking numbers have square roots that can be simplified, and that is only 37% of the numbers listed.

You may not expect it, but 1400 is one of the numbers with 24 factors. Let me tell you a little bit about 1400 and why it has so many factors:

  • 1400 is a composite number.
  • Prime factorization: 1400 = 2 × 2 × 2 × 5 × 5 × 7, which can be written 1400 = 2³ × 5² × 7
  • 1400 has at least one exponent greater than 1 in its prime factorization so √1400 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1400 = (√100)(√14) = 10√14
  • The exponents in the prime factorization are 3, 2, and 1. Adding one to each exponent and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1400 has exactly 24 factors.
  • The factors of 1400 are outlined with their factor pair partners in the graphic below.

1400 is the hypotenuse of TWO Pythagorean triples:
392-1344-1400 which is (7-24-25) times 56
840-1120-1400 which is (3-4-5) times 280

31 Flavors of 1396

The first 52 triangular numbers are 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378.

Stetson.edu informs us that 1396 can be written as the sum of three triangular numbers in 31 different ways. It is the smallest number that can make that claim!

That 31st way is written with three consecutive triangular numbers, 435, 465, and 496, which are the 29th, 30th, and 31st triangular numbers respectively. That fact makes 1396 the 31st Centered Triangular Number as well!

That is, at least, 1396 is the 31st number on the list. You can also calculate it using this formula: [3(30²) + 3(30) + 2]/2 = 1396

Here’s more about the number 1396:

  • 1396 is a composite number.
  • Prime factorization: 1396 = 2 × 2 × 349, which can be written 1396 = 2² × 349
  • 1396 has at least one exponent greater than 1 in its prime factorization so √1396 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1396 = (√4)(√349) = 2√349
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1396 has exactly 6 factors.
  • The factors of 1396 are outlined with their factor pair partners in the graphic below.

1396 is the sum of two squares:
36² + 10² = 1396

1396 is the hypotenuse of a Pythagorean triple:
720-1196-1396 calculated from 2(36)(10), 36² – 10², 36² + 10²

1395 and Level 2

1391 is the 22nd Friedman number, and there are TWO reasons why!

See! Factoring numbers can be such an exciting adventure! Can you find the factors for this puzzle?

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403\

Here’s more about the number 1395:

  • 1395 is a composite number.
  • Prime factorization: 1395 = 3 × 3 × 5 × 31, which can be written 1395 = 3² × 5 × 31
  • 1395 has at least one exponent greater than 1 in its prime factorization so √1395 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1395 = (√9)(√155) = 3√155
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1395 has exactly 12 factors.
  • The factors of 1395 are outlined with their factor pair partners in the graphic below.

You can see the reasons 1395 is the 22nd Friedman numbers in these factor pairs:
15 × 93 = 1395
45 × 31 = 5×9×31 = 1395, that one uses the digits in reverse order!

1395 is also the hypotenuse of a Pythagorean triple:
837-1116-1395 which is (3-4-5) times 279