The 153rd Playful Math Education Blog Carnival!

Welcome to the 153rd Playful Math Education Carnival! Thanks to those who blogged and/or tweeted about math, we have another fun-filled carnival this month. Since a picture is worth 1000 words, and tweets usually have lovely pictures and captions included with them, I’ve embedded a lot of tweets in this carnival. Many of the tweets include links to blog posts. You can be transported directly to any area of the carnival you desire by clicking one of the following links:

Ukraine and Math

I have been very upset about the recent events in Ukraine and wondered how I could possibly publish a cheery, playful carnival at this time.

I decided to publish the carnival but include a couple of blog posts that link math and Ukraine.

This first post is a poem about the current situation: Evil Adds Up.

The second post is about Voroinoi Diagrams and Ukrainian mathematician, Georgii Voronoi. I am glad to know a little bit about Goergii Voronoi now after reading that post.

A Fishing Pond

You can learn some fun math facts by reading blogs. A few years ago I read a post on the Math Online Tom Circle blog that made the number 153 unforgettable for me. 153 is known as the St. Peter Fish Number.

Now fishing pond booths are often a part of a traditional carnival so this 153rd edition of the Playful Math Education Blog Carnival just has to have one, too. Its fishing booth has 153 fish in it representing the 153 fish Simon Peter caught in John 21: 9-14.

I made the fish tessellate because tessellation is a cool mathematical concept. The fish form a triangle because 153 is a triangular number. I colored the fish to show that
153 = 5! + 4! + 3! + 2! + 1! I like that 5! is also a triangular number so I put it at the top of the triangle, but 1! and 3! are triangular numbers, too. Can you use addition on the graphic to show that 153 = 1³+5³+3³?

A Little Magic:

That same Tom Circle blog post also revealed the magician’s secret behind a potential math magic trick:
Pick a number, ANY number. Multiply it by 3. Then find the sum of the cubes of its digits. Find the sum of the cubes of the digits of that new number. I might have you repeat that last step a few times. I predict your final number is. . .

No matter what number you choose, I can accurately predict what your final number will be. If you open the sealed envelope in my hand, you will see that I did indeed predict your final number, 153.

Pat’s Blog teaches about another magical number property in Squares That Parrot Their Roots.

Creative Learning AfrikA+ writes about the secret of performing well on tests in It’s not Math Magic, It’s Consistency.

The Enchanted Tweeting Room

Speaking of the number 153, Jo Morgan recently published her 153rd Mathsgem post with many ideas from the Twitterverse:

In her 95th Monday Must-Reads blog post, Sara Carter shared some great ideas she saw on Twitter: a math word wall, some Desmos Gingerbread Houses, a Find-the-Imposter Spiderman Surds activity, A Polynomial Two Truths and a Lie game, and much more.

Math Art

Leonardo DeVinci and many other famous artists were also famous for their mathematics. Mathematics used to be considered a liberal art. Denise Gaskins encourages us to bring back the joy of learning math in part one of Rediscover the Liberal Art of Mathematics.

RobertLovesPi uses enneagrams, regular hexagons, and other polygons to make a lovely artistic design. He also creates a shimmering 3-D shape in A Faceted Rhombicosidocecahedron with 540 Faces.

Paula Beardell Krieg wrote about the experience of directly teaching paper folding and indirectly teaching mathematics over zoom for six weeks in A Lovely Experiment.

We can use Desmos to create stunning artwork:

Math Games

Every week Denise Gaskins shares a new game on her blog post Math Game Monday.

Julie Naturally shares some Awesome Free Math Games for Kindergartners, no electronics required.

Children at St Margaret’s Lee Church of England Primary School have been playing a domino game called the Mexican Train game. They like it so much that they’ve expressed the desire to play it at home with their families.

How much do children enjoy playing mathy board games? Just read this post by Jenorr73 of One Good Thing: Math Game Joy.


Ben Orlin of Math with Bad Drawings wrote about one of my favorite games, SET, in A Theory of Trios.

Claire Kreuz of NBC’s KARK.com blog reports that a 14-year-old High School student has developed a math game everyone can play even those with special needs.

Puzzles

Colleen Young blogs about a new publication by Jonny Griffiths in A-Level Starters.

This puzzle is my contribution to the carnival:

Here are some puzzles I found on Twitter:

Sarah Carter regularly shares puzzles that help us play with math:

 

Children’s Literature

You will want to read the responses to this next tweet. MANY biographies of mathematicians are mentioned:

Mathematical Poetry

What’s a Fib? It’s a poetry style beautifully explained by the Kitty Cats blog.

Molly Hogan of Nix the Comfort Zone wrote thoughtful poems about the Number Zero and How Many Snowflakes Were Seen out her window.

Catherine Flynn of Reading to the Core taught me about the Fibonacci style of writing poems in Poetry Fridays: Fractals, Fibonacci, and Beyond.

Count the syllables in Heidi Mordhorst of My Juicy Little Universe’s poem Jealousy and you will have counted down from nine to one.

MaryLee Hahn of A(another) Year of Reading makes us ponder our footprint in The Mathematics of Consumerism.

Number Sense

The Year One Class had a wonderful time playing with numbers as they put the numbers from 0 to 50 in their proper places on a number line and talked about number patterns.

Norah Colvin (Live, Laugh, Learn . . . Create the Possibilities blog) used easily stackable pancakes to help students have a better sense of how much 1000 is.


In Important ideas about addition, Tad Watanabe reminds us that children don’t necessarily understand concepts such as 30 being three tens. Students sometimes erroneously think of multi-digit numbers as “simply a collection of single-digit numbers that are somehow glued together.” He talks about what to do about these and a few other issues students face learning mathematics.

Jenna Laib of Embrace the Challenge observed that one of her students was having difficulty understanding negative numbers. Read what happened when she played a Tiny Number Game with her.

Multiplication and Factoring


Fractions

Geometry

Look at the pictures. You can tell that Mrs. Bracken’s class enjoyed exploring and discussing ways to display four squares and their reflections.

Check out Pat’s Blog So You Thought You Knew Everything About Equilateral Triangles.

Math History Museum

Health and First Aid Station

Math Anxiety can be a real health issue:

Math Teachers can experience a different type of Math Anxiety:


The Math Teacher Experience

What if a math lesson is fun but the concepts won’t be a major part of the end-of-year test? Pay attention to Melissa D of the Dean of Math blog post, It’s a fun unit, but it’s not really necessary.

Anna of iamamathteacher.blogspot.com shares how her school year has gone so far.

Robert Talbert discusses going from a grading-to-upgrading in an upper-level math course.

Football and Math

In Inequalities on the Gridiron, Dick Lipton and Ken Regan talk about why the Buffalo Bills weren’t in the Superbowl and whether or not the c in the inequality
a² – b² = c is positive or negative.

Check out Eric Eager’s article: Football’s lessons about mathematics, academia, and industry.

The Quillette has an interesting, although possibly controversial read: It’s Time to Start Treating High School Math Like Football.

If a math student, football player, or anyone else feels like a zero, they could benefit from Fran Carona inspiring You Are Not a Zero about Cooper Kupp, the Super Bowl MVP.

Statistics

Stanleyavestaff room 6 students enjoyed the lollipop statistics assignment so much fun that they didn’t even know they were doing math.

Joseph Nebus of Another Blog, Meanwhile made a humorous pie chart in Statistics Saturday: My Schedule for Doing Things. Many students and even adults can probably relate to it.

Calculus and Higher Math

Joseph Nebus of Nebusresearch has been reading a biography of Pierre-Simon LaPlace, so naturally, he blogs about monkeys, typewriters, and William Shakespeare in Some Progress on the Infinitude of Monkeys.

Math Wordle

Wordle has recently taken the world by storm. Got some math vocabulary words for your students? No matter how long the words are, your students can try to guess such words when they’re presented as wordles that you’ve made with the help of mywordle.strivemath.com. I made the one below. I told my son it was a math term and asked him to solve it:

There are also numerous wordles based on numbers rather than letters:

Stand-up Comedy Show


Archon’s Den shared some clever Math One-liners that will make you and your students either roll on the floor with laughter or roll your eyes.

Other Carnivals

The 201st edition of the Carnival of Mathematics can be found at the team at Ganit Charcha.

Last month the 152nd Playful Math Carnival was hosted by Denise Gaskins. Perhaps you would like to host the next carnival or one later in the year. We need more volunteers! To volunteer to host the carnival go to Denise Gaskins’ Carnival Volunteer Page.

1718 Factor Fits Valentine

Today’s Puzzle:

Happy Valentine’s Day! I hope you enjoy my Valentine to YOU!

Other Mathematical Valentines:

Colleen Young of Mathematics, Learning and Technology has a nice collection of mathematical hearts For Valentine’s Day.

Factors of 1718:

  • 1718 is a composite number.
  • Prime factorization: 1718 = 2 × 859.
  • 1718 has no exponents greater than 1 in its prime factorization, so √1718 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 1718 has exactly 4 factors.
  • The factors of 1718 are outlined with their factor pair partners in the graphic below.

More About the Number 1718:

1718 is the sum of four consecutive numbers:
428 + 429 + 430 + 431 = 1718.

 

1717 Factor Fits for Your Valentine

Today’s Puzzle:

Solve this Valentine-themed Factor Fits puzzle with both logic and heart!

Factors of 1717:

  • 1717 is a composite number.
  • Prime factorization: 1717 = 17 × 101.
  • 1717 has no exponents greater than 1 in its prime factorization, so √1717 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 1717 has exactly 4 factors.
  • The factors of 1717 are outlined with their factor pair partners in the graphic below.

More About the Number 1717:

1717 is the sum of two squares two different ways:
41² + 6² = 1717, and
39² + 14² = 1717.

1717 is the hypotenuse of FOUR Pythagorean triples:
340 1683 1717, which is 17 times (20-99-101),
492 1645 1717, calculated from 2(41)(6), 41² – 6², 41² + 6²,
808 1515 1717, which is (8-15-17) times 101, and
1092 1325 1717, calculated from 2(39)(14), 39² – 14², 39² + 14².

Two Factor Pairs of 1716 Make Sum-Difference!

Today’s Puzzle:

How many factor pairs does 1716 have? Twelve. The factors in one of those factor pairs add up to 145 and the factors in a different one subtract to 145. If you can find those two factor pairs, then you can solve this puzzle!

Try to solve it all by yourself, but if you need some help, you can scroll down to learn more about the factors of 1716.

A Factor Tree for 1716:

3 is a factor of 1716 because 1 + 7 + 1 + 6 = 15, a multiple of 3.
11 is a factor of 1716 because 1 – 7 + 1 – 6 = -11.

3 ·11 is 33. Here is one of MANY possible factor trees for 1716. This one uses the fact that 33 · 52 = 1716:

Factors of 1716:

  • 1716 is a composite number.
  • Prime factorization: 1716 = 2 × 2 × 3 × 11 × 13, which can be written 1716 = 2² × 3 × 11 × 13.
  • 1716 has at least one exponent greater than 1 in its prime factorization so √1716 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1716 = (√4)(√429) = 2√429.
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1716 has exactly 24 factors.
  • The factors of 1716 are outlined with their factor pair partners in the graphic below.

Here’s a chart of those same factor pairs but in reverse order with their sums and differences included.

More About the Number 1716:

1716 is the hypotenuse of a Pythagorean triple:
660-1584-1716, which is (5-12-13) times 132.

OEIS.org reminds us that 1716 is in the 6th and 7th columns of the 13th row of Pascal’s Triangle:

1715 A Lot More of a Subtraction Distraction

Today’s Puzzle:

Last time I published a puzzle with the last clue missing. Leaving out the first or the last clue only makes the puzzle slightly more difficult. What if I left out a clue more in the middle of the puzzle. I gave that some thought and designed today’s puzzle. I soon realized that I had to let you know that the 12 is one of the last eight boxes. There is only one solution. Can you find it?

I posted a solution video for it on Twitter:

Factors of 1715:

  • 1715 is a composite number.
  • Prime factorization: 1715 = 5 × 7 × 7 × 7, which can be written 1715 = 5 × 7³.
  • 1715 has at least one exponent greater than 1 in its prime factorization so √1715 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1715 = (√49)(√35) = 7√35.
  • The exponents in the prime factorization are 1 and 3. Adding one to each exponent and multiplying we get (1 + 1)(3 + 1) = 2 × 4 = 8. Therefore 1715 has exactly 8 factors.
  • The factors of 1715 are outlined with their factor pairs in the graphic below.

More About the number 1715:

1715 = 1·7³·1·5. Thank you, OEIS.org for that fun fact!

1715 is the hypotenuse of a Pythagorean triple:
1029-1372-1715, which is (3-4-5) times 343.

1715 = 5·7³.
5·7º = 2² + 1².
5·7¹ cannot be written as the sum of two squares.
5·7² = 14² + 7².
5·7³ cannot be written as the sum of two squares.
5·7⁴ = 98² + 49².

What do you notice? What do you wonder?

1714 Will the Factors in This Puzzle Give You Fits?

Today’s Puzzle:

12 and 24 have several common factors, but only one of them works in this puzzle. Will it be 2, 3, 4, 6, or 12?

What about 40 and 60’s common factors?

Don’t guess which factor to use. Start elsewhere in the puzzle where there’s only one possible common factor. Then use logic to eliminate some of the factor possibilities for 12, 24 and 40, 60. You will have to think, but it won’t be too difficult.

Factors of 1714:

  • 1714 is a composite number.
  • Prime factorization: 1714 = 2 × 857.
  • 1714 has no exponents greater than 1 in its prime factorization, so √1714 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 1714 has exactly 4 factors.
  • The factors of 1714 are outlined with their factor pair partners in the graphic below.

More About the Number 1714:

1714 is the sum of two squares:
33² + 25² = 1714.

1714 is the hypotenuse of a Pythagorean triple:
464-1650-1714, calculated from 33² – 25², 2(33)(25), 33² + 25².
It is also 2 times (232-825-857).

1713 A Little More of a Subtraction Distraction

Today’s Puzzle:

It occurred to me that as long as the last box is neither 1 nor 12 that I could leave the clue above it blank. Can you still solve the puzzle?

Factors of 1713:

  • 1713 is a composite number.
  • Prime factorization: 1713 = 3 × 571.
  • 1713 has no exponents greater than 1 in its prime factorization, so √1713 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 1713 has exactly 4 factors.
  • The factors of 1713 are outlined with their factor pair partners in the graphic below.

More About the Number 1713:

1713 is the difference of two squares in two different ways:
857² – 856² = 1713, and
287² – 284² = 1713.

1713 is the sum of two, three, and six consecutive numbers:
856 + 857 = 1713,
570 + 571 + 572 = 1713, and
283 + 284 + 285 + 286 + 287 + 288 = 1713.

Do you see any relationship between those two facts?

1712 Can You Make the Factors Fit?

Today’s Puzzle:

This Factor Fits puzzle starts off fairly easy before it potentially might give you fits trying to place the rest of the factors. Are you game?

Factors of 1712:

  • 1712 is a composite number.
  • Prime factorization: 1712 = 2 × 2 × 2 × 2 × 107, which can be written 1712 = 2⁴ × 107.
  • 1712 has at least one exponent greater than 1 in its prime factorization so √1712 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1712 = (√16)(√107) = 4√107.
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1712 has exactly 10 factors.
  • The factors of 1712 are outlined with their factor pair partners in the graphic below.

More About the Number 1712:

1712 is the difference of two squares in three different ways:
429² – 427² = 1712,
216² – 212² = 1712, and
111² – 103² = 1712.

1711 is a Triangular Number So The Taxman Wants His Share

Today’s Puzzle:

Imagine this puzzle is made up of 58 envelopes, each containing the amount of money printed in bold type on its front. You can’t take any of the envelopes without the Taxman also taking a share. The Taxman will take EVERY available envelope that has a factor of the number you take on it. When you have taken all the cash you can, the Taxman gets ALL the leftover cash, and the game is over. You want the Taxman to get as little money as possible.

How much money is at stake? (58 × 59)/2 = 1711. That means 1711 is a triangular number, the sum of all the numbers from 1 to 58. Thus, the total amount of money you will be splitting with the Taxman is 1711.

To play this game, you can print the cards from this excel file: Taxman & 1537-1544. The factors of a number is printed in small type at the top of its card.

Here below I show the order I selected the cards when I played. For example, I took 53, and all the Taxman got was 1. I took 49, and the only card left for the Taxman to take was 7, and so forth. The last card I could take was 42, and the Taxman got 21, but since I couldn’t take anymore cards, the Taxman also got 31, 34, 37, 41, 43, and 47.

To win the game, you must get over half of the 1711 cash, but of course, you will want the Taxman to get much less than nearly half the money.

I didn’t want to make a long addition problem to find out how much I kept, so I came up with an easier strategy: I pushed the Taxman’s share to the side, and looked for ways to make 100 from my cards. (It is almost the 100th day of school, after all.) I found 9 ways to make 100, so I clearly kept more than half of the 1711.

I added that final 53 + 50 + 39 + 36 in my head by thinking that if I took 3 away from 53 and gave it to the 36, it would be the same as 50 + 50 + 39 + 39, a fairly easy sum.

Factors of 1711:

  • 1711 is a composite number.
  • Prime factorization: 1711 = 29 × 59.
  • 1711 has no exponents greater than 1 in its prime factorization, so √1711 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 1711 has exactly 4 factors.
  • The factors of 1711 are outlined with their factor pair partners in the graphic below.

1711 is a Shape-Shifting Number:

  • 1711 is the 58th Triangular Number because (58·59)/2 = 1711.
  • It is the 20th Centered Nonagonal Number because it is one more than nine times the 19th triangular number: 9(19·20)/2 + 1 =1711. AND
  • It is the 19th Centered Decagonal Number because it is one more than 10 times the 18th triangular number: 10(18·19)/2 +1 = 1711.

That last figure I’ve illustrated below with ten triangles circled around the center dot:

Now get this: Not only is 1711 the 19th Centered Decagonal Number, but similar-looking 17111 is the 59th Centered Decagonal Number! (A mere coincidence, but the 59th is even cooler because 59·29 = 1711.)

More About the Number 1711:

1711 is the hypotenuse of a Pythagorean triple:
1180-1239-1711, which is (20-21-29) times 59.

1711 is the difference of two squares in two different ways:
856² – 855² = 1711, and
44² – 15² = 1711.

1711 is a fun number to explore!

Find the Factor Pairs of 1710. Which One Sums to 181? Which One Subtracts to 181?

Today’s Puzzle:

1710 has TWELVE factor pairs. Two of those factor pairs are special. If you add up the factors in one of them, you will get 181, but get this, if you subtract the factors in the other one, you will also get 181. Find the two factor pairs that do that, and you will have essentially solved this puzzle:

This Sum-Difference puzzle would be helpful if you had to factor any of these trinomials:
45x² – 181x + 38,
95x² – 181x – 18,
90x² + 181x + 19,
38x² + 181x – 45,

The trinomials are in the form ax² + bx + c, where b = 181, and a·c = ±1710. Whether the 1710 is positive or negative makes all the difference in the world when factoring them!

Factors of 1710:

  • 1710 is a composite number.
  • Prime factorization: 1710 = 2 × 3 × 3 × 5 × 19, which can be written 1710 = 2 × 3² × 5 × 19.
  • 1710 has at least one exponent greater than 1 in its prime factorization so √1710 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1710 = (√9)(√190) = 3√190.
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1710 has exactly 24 factors.
  • The factors of 1710 are outlined with their factor pair partners in the graphic below.

The following chart lists 1710’s factor pairs with their sums and differences:

More About the Number 1710:

1710 is the hypotenuse of a Pythagorean triple:
1026-1368-1710, which is (3-4-5) times 342.