The 120th Playful Math Carnival

Plinko is a fun carnival game of chance. This Plinko board is really just a portion of Pascal’s triangle. Stetson.edu 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 journalling 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.

I hesitate a little to share this next one. However, older students may enjoy reading a little satire from the Onion that was shared this month on the Bluebird of Bitterness, Young girls creeped out by older scientists constantly trying to lure them into STEM. It certainly could give you something to talk about.

The next stop at our carnival is a house of horrors that is simply terrifying to some people. It is known as . . . . .

Math Anxiety

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.

Alyssa lets you take a peek into the world of one who suffers from math anxiety in her post What Does Math Anxiety Look Like?

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.

Preventing and Treating Math Anxiety

So as you can see Math Anxiety is a real concern. What can you do about it? Josh Rappaport of Math Chat advises How to talk about math without scaring children off.

And of course, Denise Gaskins’ Let’s Play Math Blog is filled with ways to PLAY with math. Play can relieve a lot of anxiety. Recently Denise posted a quote by Rózsa Péter about math being worthy of our time and how Rózsa’s class of twelve-year-old girls begged her to let them explore the Euclidean algorithm. These girls felt no fear; it was a joyful experience for them the entire time.

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.

One great way to make mathematical art is to use mirrors as demonstrated in these photos by Annie Fetter when she went to Math on a Stick.

Robert Loves Pi once again has created some beautiful, rotating 3-dimensional mathematical art for us all to enjoy.

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?

I took a photo at a Hungarian museum village and turned it into a mathematical puzzle/lesson for young ones by asking a couple of simple questions. How Are They the Same? How are They Different?

Logic

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.

Science Book A Day reviewed mathematician Eugenia Cheng’s book, The Art of Logic: How to Make Sense in a World That Doesn’t.

Math Literature and Books

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.

Crafts, Fashion, Souvenir booths

At this next carnival booth, you can buy a variety of clothing items. Should you buy any of them? Fashion Math-Thinking about the Cost Per Wear shares a formula created to help you make that decision.

TerifiCreations by Teri Lewis asks, “Has anyone ever written an article encouraging quilters to do math?” If any quilters out there struggle with the math, she will gladly help out.

Mathemagic or Carni Game?

How to get super-rich; millionaire math suggests 13 different ways to get to a million and would be a fun way to increase number sense for students who already know how to multiply.

Sometimes students come up with ridiculous answers to word problems. DC Gilbert shares a disastrous story and concludes, “Mathematics! It is Really That Important!

Using statistics to tell lies: Open Mind gives an example in USA Temperatures: Can I Sucker You?

Winning Mathematical Game Skills

The son of one of Math Mammoth’s customers created a flash program that helps second-grade students practice simple addition and subtraction facts. Skills require practice so check it out!

Resourceaholic offers some fun beginning-of-the-school-year activities for year 7 students.

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.

If you’re teaching the Fundamental Counting Principle, I’m sure you can find a way to use Wrong Hands’ clever/funny comic Lesser super-hero movie title generator.

Chris McMullen can answer your students’ question, Which Calculus Skills are most essential, practical?

How do you prove that e is an irrational number? Mjlawler tackles that problem in Walking through the proof that e is irrational with a kid.

The Carnival of the Future

Joseph Nebus, who will host the carnival in September at his blog, NebusResearch mentioned some comics that could lessen geometry anxiety in Reading the Comics, Ragged Ends Edition.

Joseph also writes a humor blog that sometimes has gems like the Venn Diagram he made for his post Statistics Saturday: Trivia Night Questions, by Kind.

I can tell that Joseph is pretty pumped about writing the carnival next month. Read The Mathematics Carnival is coming! and enjoy his enthusiasm.

You can also enjoy the August 2018 edition of the Carnival of Mathematics.

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!

 

 

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1198 Challenge Puzzle

You can solve this Find the Factors 1 – 10 puzzle if you use logic. Guessing and checking will likely only frustrate you. Go ahead and give logic a try!

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share some facts about the number 1198:

  • 1198 is a composite number.
  • Prime factorization: 1198 = 2 × 599
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1198 has exactly 4 factors.
  • Factors of 1198: 1, 2, 599, 1198
  • Factor pairs: 1198 = 1 × 1198 or 2 × 599
  • 1198 has no square factors that allow its square root to be simplified. √1198 ≈ 34.61214

1198 is also palindrome 262 in BASE 23

1197 Mystery Level

The first few moves needed to solve this puzzle might not be too hard, but soon enough it might get a bit tougher. Nevertheless, its one solution can be found using logic and an ordinary 12×12 multiplication table.

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Here are facts about the number 1197:

  • 1197 is a composite number.
  • Prime factorization: 1197 = 3 × 3 × 7 × 19, which can be written 1197 = 3² × 7 × 19
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1197 has exactly 12 factors.
  • Factors of 1197: 1, 3, 7, 9, 19, 21, 57, 63, 133, 171, 399, 1197
  • Factor pairs: 1197 = 1 × 1197, 3 × 399, 7 × 171, 9 × 133, 19 × 63, or 21 × 57
  • Taking the factor pair with the largest square number factor, we get √1197 = (√9)(√133) = 3√133 ≈ 34.59769

1197 is the sum of these eleven consecutive prime numbers:
83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1197

1197 looks interesting to me when it is written in some other bases:
It’s 3330 in BASE 7 because 3(7³ + 7² + 7¹) = 3(399) = 1197,
and it’s 2255 in BASE 8.
It’s 999 in BASE 11, because 9(11² + 11 + 1) = 9(133) = 1197,
and it’s 1K1 in BASE 26 (K is 20 base 10)

 

1196 and Level 6

In this puzzle, the permissible common factors of 48 and 72 are 6, 8, and 12. For clues 8 and 16, you can choose from common factors 2, 4, or 8. Which choices will make the puzzle work? I’m not telling, but I promise that the entire puzzle can be solved using logic and a basic knowledge of a 12×12 multiplication table. There is only one solution.

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Here are some facts about the number 1196:

  • 1196 is a composite number.
  • Prime factorization: 1196 = 2 × 2 × 13 × 23, which can be written 1196 = 2² × 13 × 23
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1196 has exactly 12 factors.
  • Factors of 1196: 1, 2, 4, 13, 23, 26, 46, 52, 92, 299, 598, 1196
  • Factor pairs: 1196 = 1 × 1196, 2 × 598, 4 × 299, 13 × 92, 23 × 52, or 26 × 46
  • Taking the factor pair with the largest square number factor, we get √1196 = (√4)(√299) = 2√299 ≈ 34.58323

1196 is the hypotenuse of a Pythagorean triple:
460-1104-1196 which is (5-12-13) times 92

1196 is a palindrome in three different bases:
It’s 14241 in BASE 5,
838 in BASE 12, and
616 in BASE 14

1195 You Can Find the Answer in This Book

The new school year is underway. Much may have been forgotten over the summer. If you don’t quite remember all the multiplication tables, this puzzle book can help you remember them AND help your brain grow. You might still find it a challenge, but that only makes it more fun!

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share a few facts about the number 1195:

  • 1195 is a composite number.
  • Prime factorization: 1195 = 5 × 239
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1195 has exactly 4 factors.
  • Factors of 1195: 1, 5, 239, 1195
  • Factor pairs: 1195 = 1 × 1195 or 5 × 239
  • 1195 has no square factors that allow its square root to be simplified. √1195 ≈ 34.56877

1195 is also the hypotenuse of a Pythagorean triple:
717-956-1195 which is (3-4-5) times 239

1194 and Level 4

The more multiplication facts you know, the easier these puzzles become. Working on these puzzles can help you learn the multiplication table better. Go ahead,  give this puzzle a try!

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Here are a few facts about the number 1194:

  • 1194 is a composite number.
  • Prime factorization: 1194 = 2 × 3 × 199
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1194 has exactly 8 factors.
  • Factors of 1194: 1, 2, 3, 6, 199, 398, 597, 1194
  • Factor pairs: 1194 = 1 × 1194, 2 × 597, 3 × 398, or 6 × 199
  • 1194 has no square factors that allow its square root to be simplified. √1194 ≈ 34.5543

1194 is the sum of consecutive prime numbers two ways:
131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 = 1194
283 + 293 + 307 + 311 = 1194

1194 is palindrome 424 in BASE 17

 

1193 Math Carnival Games

During the last week of every month, there is a math education blog carnival happening somewhere in the blogosphere. This month it will happen on my blog! Why do I get to host it? I sent an email to Denise Gaskins who coordinates the carnival and requested the privilege. If you would like to host it in the future, let her know.

In the meantime, you can help me with my carnival. Math can be so much fun for kids from preschool age and even all the way up to high school. If you blog about how that, I would love to include one or more of your posts in my carnival. You’ve poured your heart and soul into your posts, so why not promote it at no cost to you?  Don’t be shy! I want to read it, and other people will want to read it, too.

The deadline for submitting posts to my carnival is Friday, August 24th. There is a form for you to submit a link to your post on Denise Gaskins website. Then the following week you will be able to enjoy the carnival even more because of your participation!

Now it will be my pleasure to tell you a few facts about the number 1193:

  • 1193 is a prime number.
  • Prime factorization: 1193 is prime.
  • The exponent of prime number 1193 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1193 has exactly 2 factors.
  • Factors of 1193: 1, 1193
  • Factor pairs: 1193 = 1 × 1193
  • 1193 has no square factors that allow its square root to be simplified. √1193 ≈ 34.53983

How do we know that 1193 is a prime number? If 1193 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1193 ≈ 34.5. Since 1193 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1193 is a prime number.

1193 is the sum of five consecutive prime numbers:
229 + 233 + 239 + 241 + 251 = 1193

32² + 13² = 1193

1193 is the hypotenuse of a Pythagorean triple:
832-855-1193 calculated from 2(32)(13), 32² – 13², 32² + 13²

Here’s another way we know that 1193 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 13² = 1193 with 32 and 13 having no common prime factors, 1193 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1193 ≈ 34.5. Since 1193 is not divisible by 5, 13, 17, or 29, we know that 1193 is a prime number.

1192 Experiential Learning Helps Students Understand Their Capacity

Sara Van Der Werf inspired me last year with her post titled Why You Need a PLAY TABLE in your mathematics classroom ASAP. I’ve seen merit in using play to teach and learn mathematics for some time now. What Sara’s post did was make me long to be back in a mathematics classroom so I could have such a play table!

Last week I started working at the American Academy of Innovation, a charter school near that engages students in learning through projects and other activities.

Each member of the faculty was given the assignment to make something that reflects his or her teaching philosophy. Here’s the one I put together:

On my display, I included the words, “Experiential Learning Helps Students Understand Their Capacity.” That’s my teaching philosophy.

The wording I used was intentional.

  •  “Their” might refer to the students or to the geometric solids.
  • “Capacity” has two appropriate definitions stated so perfectly by Google in the screenshot below:
    1) the ability or power the students “have to do, experience, or understand something”, and 2) the maximum amount the geometric solids can contain.

The scattered rice symbolizes that these learning experiences can sometimes be messy. We will learn more if we aren’t afraid to make mistakes or a little mess.

I am so excited! I’ve taken all my math toys out of storage and will have an ever-changing play table in my classroom where students can play and learn mathematics. Thank you, Sara Van Der Werf for inspiring me!

Now I’ll write a little about the number 1192:

  • 1192 is a composite number.
  • Prime factorization: 1192 = 2 × 2 × 2 × 149, which can be written 1192 = 2³ × 149
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1192 has exactly 8 factors.
  • Factors of 1192: 1, 2, 4, 8, 149, 298, 596, 1192
  • Factor pairs: 1192 = 1 × 1192, 2 × 596, 4 × 298, or 8 × 149
  • Taking the factor pair with the largest square number factor, we get √1192 = (√4)(√298) = 2√298 ≈ 34.52535

1192 is the sum of two consecutive prime numbers:
593 + 599 = 1192

34² +  6² = 1192

1192 is the hypotenuse of a Pythagorean triple:
408-1120-1192 calculated from 2(34)( 6), 34² –  6², 34² +  6²
It is also 8 times (51-140-149)

1192 is a cool-looking 3322 in BASE 7

1191 Happy Birthday, Brent

Today I’m saying “Happy Birthday” to my son, Brent with a yummy but mysterious puzzle cake. I’ve made the puzzle just a little bit harder than normal.

Adding 13 to the puzzle actually makes it easier to solve, but adding 14 makes some multiples of 7 more difficult. For example, the allowable common factors of 70 and 35 are now 7 AND 5, and the allowable common factors of 28 and 56 are now 4, 7, and 14.

As always there is only one solution. I know my son can solve this puzzle, Can you?

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share some facts about the number 1191:

  • 1191 is a composite number.
  • Prime factorization: 1191 = 3 × 397
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1191 has exactly 4 factors.
  • Factors of 1191: 1, 3, 397, 1191
  • Factor pairs: 1191 = 1 × 1191 or 3 × 397
  • 1191 has no square factors that allow its square root to be simplified. √1191 ≈ 34.51087

1191 is the hypotenuse of a Pythagorean triple:
684-975-1191 which is 3 times (228-325-397)

1191 is repdigit 111 in BASE 34 because
34² + 34 + 1 = 35(34) + 1 = 1191

1190 and Level 3

The common factors of 108 and 120 are 1, 2, 3, 4, 6, and 12, but pick the one that will only put numbers from 1 to 12 in the first column. Then work down that first column cell by cell writing in the factors of the clues as you go. Each number from 1 to 12 must go somewhere in both the first column and the top row.

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share some information about the number 1190:

  • 1190 is a composite number.
  • Prime factorization: 1190 = 2 × 5 × 7 × 17
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 110. Therefore 1190 has exactly 110 factors.
  • Factors of 1190: 1, 2, 5, 7, 10, 14, 17, 34, 35, 70, 85, 119, 170, 238, 595, 1190
  • Factor pairs: 1190 = 1 × 1190, 2 × 595, 5 × 238, 7 × 170, 10 × 119, 14 × 85, 17 × 70, or 34 × 35
  • 1190 has no square factors that allow its square root to be simplified. √1190 ≈ 34.49638

Because 1190 is the product of consecutive numbers 34 and 35, we know it is the sum of the first 34 EVEN numbers. Instead of writing all of those 34 numbers, we can use a some mathematical shorthand and simply write:
2 + 4 + 6 + 8 + . . . + 64 + 66 + 68 = 1190

1190 is the hypotenuse of FOUR Pythagorean triples:
182-1176-1190 which is 14 times (13-84-85)
504-1078-1190 which is 14 times (36-77-85)
560-1050-1190 which is (8-15-17) times 70
714-952-1190 which is (3-4-5) times 238

I like the way 1190 looks in some other bases:
It’s 707 in BASE 13 because 7(13²) + 7(1) = 7(169 + 1) = 7(170) = 1190,
545 in BASE 15,
2A2 in BASE 22,
1C1 in BASE 29 (C is 12 base 10),
and Y0 in BASE 35 (Y is 34 base 10) because 34(35) = 1190