What Kind of Shape is 1520 in?

Today’s Puzzle:

Sure, it’s a rectangle with whole-number sides in 10 different ways, but what kind of REGULAR polygonal shape can 1520 be made into? I will tell you that the measurement of each of its sides is 32.

And thus, it is the 32nd shape of its kind. By the way, I really like how all the 32nd figurate numbers relate to each other:

We see in the chart that 1520 dots can be arranged into a pentagon. Just how do we do that? Here’s how:

Do you see from the graphic that 1520 is 32 more than three times the 31st triangular number?

1520 is also related to triangular numbers in another way: Today I learned that all pentagonal numbers are 1/3 of a triangular number.  Indeed, 1520 is 1/3 of the 95th triangular number:
(1/3) of (95)(96)/2 = 1520.

Pretty cool, I think!

Factors of 1520:

  • 1520 is a composite number.
  • Prime factorization: 1520 = 2 × 2 × 2 × 2 × 5 × 19, which can be written 1520 = 2⁴ × 5 × 19
  • 1520 has at least one exponent greater than 1 in its prime factorization so √1520 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1520 = (√16)(√95) = 4√95. The exponents in the prime factorization are 4, 1, and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1520 has exactly 20 factors.
  • The factors of 1520 are outlined with their factor pair partners in the graphic below.

More about the Number 1520:

Recently someone on twitter asked:

If you look at the whole thread, you will see how a few people explained this important concept using arrays. Here is my attempt to explain the difference of two squares using 1520 and arrays:

As I mentioned before, 1520 has 10 rectangles with whole-number sides. The one with the smallest perimeter is 38 × 40, and it is the easiest to use to demonstrate how 1520 is the difference of two squares:

1520 Difference of Two Squares

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I made that gif be as slow as I could without duplicating any of the frames, but it still goes pretty fast.

1520 is, in fact, the difference of two squares in six different ways:
39² – 1² = 1520,
48² – 28² = 1520,
81² – 71² = 1520,
99² – 91² = 1520,
192² – 188² = 1520, and
381² – 379² = 1520.

1520 is also the hypotenuse of a Pythagorean triple:
912-1216-1520, which is (3-4-5) times 304.

 

 

 

What Kind of Shape is 1519 in?

Today’s Puzzle:

1519, 1520, and 1521 are all figurate numbers. What kind of shape can you arrange 1519 tiny dots?

1519 is the 23rd centered hexagonal number because 23³ – 22³ = 1519.

It is also the 23rd centered hexagonal number because it is one more than six times the 23rd triangular number. Do you see the 23rd triangular number six times in the graphic above?

Factors of 1519:

  • 1519 is a composite number.
  • Prime factorization: 1519 = 7 × 7 × 31, which can be written 1519 = 7² × 31
  • 1519 has at least one exponent greater than 1 in its prime factorization so √1519 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1519 = (√49)(√31) = 7√31
  • 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 1519 has exactly 6 factors.
  • The factors of 1519 are outlined with their factor pair partners in the graphic below.

1513 is the Sum of Squares

Today’s Puzzle:

How can you arrange 1513 dots into a perfect square when √1513 is irrational?

The answer is you arrange the dots into a centered square like this:

You can arrange them like that because 1513 is the sum of consecutive squares.

Factors of 1513:

  • 1513 is a composite number.
  • Prime factorization: 1513 = 17 × 89
  • 1513 has no exponents greater than 1 in its prime factorization, so √1513 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 1513 has exactly 4 factors.
  • The factors of 1513 are outlined with their factor pair partners in the graphic below.

More about the Number 1513:

1513 is the sum of two squares in two different ways:
28² + 27²  = 1513, and
37² + 12²  = 1513.

1513 is the hypotenuse of FOUR Pythagorean triples:
55-1512-1513,  calculated from 28² – 27², 2(28)( 27), 28² + 27²
663-1360-1513, which is 17 times (39-80-89)
712-1335-1513, which is (8-15-17) times 89
888-1225-1513, calculated from 2(37)(12), 37² – 12², 37² + 12²

Could 1513 be a prime number?

Since its last two digits divided by 4 leave a remainder of 1, and 28² + 27² = 1513 with   28 and 27 having no common prime factors, 1513 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1513. Is 1513 divisible by 5, 13, 17, 29, or 37? Yes, it is divisible by 17, so 1513 is NOT a prime number.

37² + 12²  = 1513 and 37 and 12 have no common prime factors, so we could have arrived at the same result using those numbers.

Note: Numbers that are the sum of two squares in two or more ways are never prime.

 

Welcome to the 140th Playful Math Education Blog Carnival!

What’s Special about 140?

Ladies and Gentlemen, welcome to the 140th Playful Math Education Blog Carnival! Feast your eyes on the number, 140, the 7th member of the famous Square Pyramidal Number family. Watch as 140 performs these amazing feats:

140 is the 7th square pyramidal number because
(7³/3) +(7²/2) + (7/6) = 140.

140 has twelve factors and will now use them to make a lovely factor rainbow:

For140’s next trick, see what happens when it is divided by six of its non-factors:

Finally, 140 is the fourth harmonic divisor number, and Wolfram Math World even uses 140 to explain what a harmonic divisor number is.

Now let’s move on to the blog entries for this month’s carnival:

Children’s Literature and Math

Kelly Darke of Math Book Magic wrote a post about a brand new entry in children’s literature, The Boy Who Dreamed of Infinity.  This is the story of the great Indian mathematician, Ramanujan. The book is available for us to read now, but Kelly was able to read it for the first time last year. I felt so much joy inside of me as I read first her reaction to the description of the book and later to the book itself. I am grateful that the mere idea of that book prompted Kelly to create a blog to share the magic of good mathematical children’s literature.

Rhapsody in Books Weblog tells us about, Raye Montague, an African-American girl born in 1935. She was told repeatedly that her race and her gender would prevent her from becoming the mathematician that she dreamed of becoming. She didn’t heed her naysayers. The Girl with a Mind for Math: The Story of Raye Montague tells her inspiring story.

On world Tessellation Day, TheKittyCats blog introduced us to Tessellations!  Children will enjoy looking at the illustrations in the book and won’t even realize they are learning some math in the process unless someone shares that secret.

Games and Math

Alan Parr of Established 1962 explains how to play Dotty Six, a game played with a tic tac toe grid and a die. Because he asked himself, “What If Not?”, he was able to suggest some mathematically interesting variations of the game as well.

Mrs. O’Brien started her blog, Math Epiphany, very recently-in May 2020. She has written two posts about using games to learn mathematics. She writes about card games in Summer Math Fun: Math Games, and about board games in Summer Math Fun: Good Old Fashioned Games.

Denise Gaskins shares a math game that lets children complexify expressions or equations rather than simplify them. Making the expression or equation a little more complex than it was before can be great fun and wonderfully educational. Check out The Best Math Game Ever!

Do you know how to play Dara, Five Field Kono, Mu Torere, Pong Hau K’I, Shisima, or Triangle Peg Solitaire? I’ve never even heard of these games before. Mark Chubb of Thinking Mathematically introduces all of these games in Math Games-building a foundation for mathematical reasoning. One of the games is illustrated in the tweet below.

Math and Paper-Folding

How is making a pouch out of a newspaper doing math? Paula Krieg and John Golden explain that it can be more than simply taking measurements and using rulers in Pouch: Something from Almost Nothing #3.

Did you know that paper folding can help kids understand systems of linear equations?

Or paper folding can even help kids understand Knot Theory or Topology?

Math and Art

What creatures can be seen in these numbers? Click on the video to find out what one artist saw!

We can always count on Robert Loves Pi to create dazzling and beautiful geometric art. Here he tells us how relaxing it is to produce it:

Photographer Ming Thein shares 13 discussion-evoking photos in Photo essay: The texture of geometry. I’d love to hear some of those discussions!

Quilts can be stunning examples of mathematical art. Aby Dolinger of Abyquilts has created a quilt pattern she calls “Math Whiz,” and this mathy quilt was featured on the July/August 2020 COVER of Quiltmaker magazine! Congratulations Aby!

Geometry and Trigonometry

Laura of Mathsux² has written an explanation and created a video to take the mystery out of trigonometric ratios in How to Use SOHCAHTOA.

Jo Morgan’s website is filled with resources to make teaching and learning math more effective, and yet she always finds room for more ideas! She recently created her 133rd Maths Gems Post that included some playful ways to look at angles in a circle and areas of rectangles inside of a larger rectangle. . . Now to segway into word problems. . . If Jo writes two Maths Gems a month and her July 30th post was number 133, and the Playful Math Blog Carnival comes out ten times a year and this is the 140th post, when will the Maths Gems number and the Playful Math Blog number be the same number?

Word Problems

I love the giant Sequoia Trees. This blog post has story problem suggestions about Sequoia Trees for every age group:

A-Hundred-Years-Ago Blog explores some Hundred-year-old Food-related Math Problems when large oranges were only 60 cents a dozen. Go back in time and enjoy solving these with your students! Let them compare them with word problems from the 21st century.

Puzzles

Alan Paar of established1962 shares his last experience helping kids play with math before the lockdown. It was a series of puzzles that made A Lesson That Will Stay With Me. He has used these adventures for 30 years and kids enjoy them so much. He was especially glad that these puzzle adventures will be their last memories of attending primary school. They was so much better than Key Stage 2 SATs.

The Find the Factors puzzles I create are a playful way to get to know the multiplication table better. This level 3 puzzle can be solved by considering the factors of 30 and 90 where only factor pairs with numbers from 1 to 10 are used. After those factors are found, write them in the appropriate cells then work your way down the puzzle row by row using logic until all of the factors are found.

Singapore Maths Tuition shares an “average” math puzzle with a twist that might baffle all but those kids who enjoy math but find little challenge in traditional math work. No worries for the rest of us; a good explanation is also included.

Mathematical Humor

Emily’s Post tells a timely math joke about three ducks that will teach while it delights children in Modern Math.

Joseph Nebus has a humor blog in which he wrote a humorous post he titled What your Favorite Polygon Says about You. I’m not sure what my favorite polygon is, but I will carefully consider all the possibilities.

The Bored Side of the Phone shares a couple of stand-up-comedy-worthy jokes about Mathematics in The Truth About Maths.

Numbers

Natural Numbers:

The counting numbers/whole number set has been further categorized! Can you imagine how? Read all about it in Publications de BOULAY’s New Whole Numbers classification. As you learned about the set of ultimate numbers, how well did your imagination serve you?

Rational and Irrational Numbers: Mike of Mike’s Math Page gives us a front-row seat observing how he teaches his sons about mathematics in Sharing John Urschel’s great video on rational and irrational numbers with my son.

Imaginary Numbers: Every year Joseph Nebus lets his readers chose mathematical topics for each letter of the alphabet, and then he writes a post about each of those A to Z topics. For 2020 he wrote a serious essay on imaginary numbers that playfully included some comics about some numbers that you can imagine.

Life skills and Math

Ladybugs or ladybirds want to teach you some math concepts. Come out in the garden with the lesson plans provided by DogwoodDays in Garden Schooling: Ladybird Maths and see what you learn!

Cooking is an important life skill and a fun way to learn about fractions and other math concepts. The For-Health blog featured a post kids and adults can enjoy together: How to Cut Down Recipes: What’s Half of ½ cup, ¾ cup, 2/3 cup and More. Verifying the given measurement equivalents can be great fun for kids so do let that happen!

How can we make our lives be as well-balanced as an equation? That’s a good question for high school students to consider. A life coach’s advice on how to find success in life is given using mathematical symbols and vocabulary in Mathematics of Life, Learn from Math symbols.

Corona Virus Math

In Wheel of Theodorus – Distance Learning Edition, MrJoyce180’s shares his students’ work creating their own, and I do mean their own, Wheels of Theodorus. All of this creating occurred virtually during the lockdown. He shares both the successes and the failures. This was one of my favorite discoveries while I created this carnival.

I didn’t have my glasses on questions Cosco’s mathematical reasoning of cake buying and serving in Let (a few of) them eat cake! Can you formulate a word problem from this post?

Statistics

When Disney produces a direct to video sequel, will a Roman numeral, an Arabic numeral, or neither most likely appear in the title? Even young children will be able to explore that topic with Joseph Nebus in this Statistics Saturday Post.

World Affairs uses cleverly represented graphs to help us understand The Math of How China Surpasses USA in 5 Years. Understanding the math behind the graphs could help us improve our situation.

Poetry and Math

Beginning with irrational numbers, Prerna’s Blog uses mathematical and poetic language to describe the Mathematics of My Mind.

Math+Life connects math with life by writing poetry. After you read Set in Stone the mathematics of sets is explained followed by how they relate to life. Do we place limits on children or adults when we categorize them into sets of different types of people?

Making Math More Inclusive

Please read Sunil Singh’s powerful and thought-provoking post, How to Begin Bringing Rich and Inclusive Math History Resources Inside K to 12 Classrooms.

I am also pleased to introduce you to the brand-new BlackWomenRockMath Blog. Their first post is The Brilliance Hiding in Plain Sight in which three women share their sobering math stories. Thankfully, they each were able to overcome negative early experiences in learning mathematics to make worthwhile contributions to mathematics education today.

Mathematics Carnivals and Amusements:

Every Monday Denise Gaskins invites you over for a Morning Coffee. There she will direct you to other mathematics blogs for your edification and amusement.

There is also a Carnival of Mathematics that may interest you. The current (184th) Carnival is hosted at Tom Rocks Maths.

I really liked putting this month’s carnival together, and I hope you have enjoyed reading it as well. Feel free to stop by and hang out whenever you’d like.

The previous Playful Math Education Blog Carnival #139 was hosted by Math Mama Writes. Be sure to check it out if you haven’t already.

I am already looking forward to the next Playful Math Education Blog Carnival which will be hosted by Joseph Nebus of Nebus Research.

Perhaps you would like to volunteer to host one of the carnivals? Contact Denise Gaskins to get on the carnival calendar! I can’t wait to see what you put together!

1505 and Level 4

Today’s Puzzle:

If you’ve never done a level 4 puzzle before, this one is a great choice. It has fewer tricky clues than usual. Start with a row or column that has two clues. Use logic and multiplication facts to figure out where to put the factors 1 to 10 in both the first column and the top row so that the given clues are the products of those factors. Have fun!

Factors of 1505:

  • 1505 is a composite number.
  • Prime factorization: 1505 = 5 × 7 × 43.
  • 1505 has no exponents greater than 1 in its prime factorization, so √1505 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 1505 has exactly 8 factors.
  • The factors of 1505 are outlined with their factor pair partners in the graphic below.

More about the Number 1505:

1505 has four different factor pairs. The numbers in each factor pair add up to an even number, so 1505 is the difference of two squares in four different ways. Here’s the complete list of those ways:
753² – 752² = 1505
153² – 148² = 1505
111² – 104² = 1505
39² – 4² = 1505

Why is 1503 a Friedman Number?

Friedman Puzzle:

Can you find an expression equaling 1503 that uses 1, 5, 0, and 3 each exactly once, but in any order, and some combination of  +, -, ×, or ÷? For this particular Friedman puzzle, none of those digits are exponents. If you can solve this Friedman puzzle, you will know why 1503 is the 24th Friedman number. You can find the solution hidden someplace in this post. (By the way, another permutation of those digits, 1530, will be the 25th Friedman number!)

Find the Factors Puzzle:

There are 14 clues in this level 2 puzzle. Use those clues and logic to place the factors 1 to 10 in both the first column and the top row. That’s how you start to turn this puzzle into a multiplication table!

Factors of 1503:

  • 1503 is a composite number.
  • Prime factorization: 1503 = 3 × 3 × 167, which can be written 1503 = 3² × 167
  • 1503 has at least one exponent greater than 1 in its prime factorization so √1503 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1503 = (√9)(√167) = 3√167
  • 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 1503 has exactly 6 factors.
  • The factors of 1503 are outlined with their factor pair partners in the graphic below.

Did you see the solution to the Friedman puzzle in that factor pair chart?

1501 is a Centered Pentagonal Number

1501 Tiny Dot Puzzle

A long time ago before calculators or computers existed, someone determined that 1501 tiny dots could be formed into a pentagon. That was a remarkable puzzle to complete! What kind of pentagon will 1501 tiny dots make?

Since 5(24)(25)/2 + 1 = 1501, it is the 25th centered pentagonal number.  Notice in the graphic below that those 1501 tiny dots can also be divided into 5 equally-sized triangles with just the center dot leftover.

Factors of 1501:

  • 1501 is a composite number.
  • Prime factorization: 1501 = 19 × 79
  • 1501 has no exponents greater than 1 in its prime factorization, so √1501 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 1501 has exactly 4 factors.
  • The factors of 1501 are outlined with their factor pair partners in the graphic below.

Another fact about the number 1501:

1501 is the difference of two squares in two different ways:
751² – 750² = 1501,
49² – 30² = 1501.

Celebrating 1500 with a Horse Race and Much More!

Pick Your Pony!

Writing 1500 posts is quite a milestone. I’ll begin the celebration with an exciting horse race! Let me explain:

Each prime number has exactly 2 factors. Every composite number between 1401 and 1500 has somewhere between 4 and 36 factors. Which quantity of factors do you think will appear most often for these numbers? Pick that amount as your pony and see how far it gets in this horse race!
1401 to 1500 Horse Race

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Did the race results surprise you? They surprised me!

Prime Factorization for numbers from 1401 to 1500:

Here’s a chart showing the prime factorization of all those numbers and the amount of factors each number has. Numbers in pink have exponents in their prime factorization so their square roots can be simplified:

Today’s Puzzles:

Let’s continue the celebration with a puzzle: 1500 has 12 different factor pairs. One of those pairs adds up to 85 and one of them subtracts to give 85. Can you find those factor pairs that make sum-difference and write them in the puzzle? You can look at all of the factor pairs of 1500 in the graphic after the puzzle, but the second puzzle is really just the first puzzle in disguise. So try solving that easier puzzle first.

Factors of 1500:

  • 1500 is a composite number.
  • Prime factorization: 1500 = 2 × 2 × 3 × 5 × 5 × 5, which can be written 1500 = 2² × 3 × 5³
  • 1500 has at least one exponent greater than 1 in its prime factorization so √1500 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1500 = (√100)(√15) = 10√15
  • The exponents in the prime factorization are 2, 1, and 3. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(3 + 1) = 3 × 2 × 4 = 24. Therefore 1500 has exactly 24 factors.
  • The factors of 1500 are outlined with their factor pair partners in the graphic below.

A Forest of Factor Trees:

Take a few minutes to hike in this forest featuring a few of the MANY possible factor trees for 1500. Celebrate each tree’s uniqueness!

Other Facts to Celebrate about 1500:

Oeis.org tells us that  (5+1) × (5+5) × (5+0) × (5+0) = 1500.

1500 is the hypotenuse of THREE Pythagorean triples:
420-1440-1500, which is (7-24-25) times 60,
528-1404-1500, which is 12 times (44-117-225),
900-1200-1500, which is (3-4-5) times 300.

1496 is a Square Pyramidal Number

Visualize 1496 Blocks:

I hoped to make a graphic illustrating that 1496 is the 16th square pyramidal number. I am thrilled that I succeeded!

Factors of 1496:

Knowing the multiplication table and some divisibility tricks helped me find some of 1496’s factors:
96 is 8 × 12, and 4 is even, so 1496 is divisible by 8.
1 – 4 + 9 – 6 = 0, so 1496 is divisible by 11.

  • 1496 is a composite number.
  • Prime factorization: 1496 = 2 × 2 × 2 × 11 × 17, which can be written 1496 = 2³ × 11 × 17
  • 1496 has at least one exponent greater than 1 in its prime factorization so √1496 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1496 = (√4)(√374) = 2√374
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1496 has exactly 16 factors.
  • The factors of 1496 are outlined with their factor pair partners in the graphic below.

Another fact about the number 1496:

1496 is the hypotenuse of a Pythagorean triple:
704-1320-1496, which is (8-15-17) times 88.

1488 Chalk Art

Geometric Chalk Art:

My grandchildren and their mom laid down some masking tape on a sidewalk panel to create lots of polygons. Then they colored it in beautifully with chalk before they removed most of the masking tape. My sidewalk never looked so good!

Perhaps you would enjoy creating some mathematical chalk art on your sidewalk, too!

Factors of 1488:

  • 1488 is a composite number.
  • Prime factorization: 1488 = 2 × 2 × 2 × 2 × 3 × 31, which can be written 1488 = 2⁴ × 3 × 31
  • 1488 has at least one exponent greater than 1 in its prime factorization so √1488 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1488 = (√16)(√93) = 4√93. The exponents in the prime factorization are 4, 1, and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1488 has exactly 20 factors.
  • The factors of 1488 are outlined with their factor pair partners in the graphic below.

More about the Number 1488:

1488 is the difference of two squares in six different ways:
373² – 371² = 1488
188² – 184² = 1488
127² – 121² = 1488
97² – 89² = 1488
68² – 56² = 1488
43² – 19² = 1488