902 and Level 5

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902 is the hypotenuse of a Pythagorean triple:

198-880-902 which is 22 times (9-40-41)

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 902 is a composite number.
  • Prime factorization: 902 = 2 × 11 × 41
  • 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 902 has exactly 8 factors.
  • Factors of 902: 1, 2, 11, 22, 41, 82, 451, 902
  • Factor pairs: 902 = 1 × 902, 2 × 451, 11 × 82, or 22 × 41
  • 902 has no square factors that allow its square root to be simplified. √902 ≈ 30.0333.

901 and Level 4

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Print the puzzles or type the solution in this excel file: 10-factors-897-904

901 is the 25th centered triangular number because (23×24 + 24×25 + 25×26)/2 = 901. That is the same as saying that 901 is the sum of the 23rd, the 24th, and the 25th triangular numbers.

901 is the sum of two squares two different ways:

  • 30² + 1² = 901
  • 26² + 15² = 901

901 is the hypotenuse of FOUR Pythagorean triples:

  • 60-899-901, calculated from 2(30)(1), 30² – 1², 30² + 1²
  • 424-795-901, which is (8-15-17) times 53
  • 451-780-901, calculated from 26² – 15², 2(26)(15), 26² + 15²
  • 476-765-901, which is 17 times (28-45-53)

Two of those were primitives. That can only happen because ALL of 901’s prime factors are Pythagorean triple hypotenuses.

  • 901 is a composite number.
  • Prime factorization: 901 = 17 × 53
  • 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 901 has exactly 4 factors.
  • Factors of 901: 1, 17, 53, 901
  • Factor pairs: 901 = 1 × 901 or 17 × 53
  • 901 has no square factors that allow its square root to be simplified. √901 ≈ 30.016662

900 Pick Your Pony. Who’ll Win This Amount of Factors Horse Race?

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I really like this rhyme that I saw for the first time this week (even though it’s all over the net):

Hey diddle diddle, the median’s the middle,
You add then divide for the mean.
The mode is the one that appears the most,
And the range is the difference between.

All of the numbers from 801 to 900 have at least 2 factors, but no more than 32 factors. 32 – 2 = 30, so 30 is the range of the amount of factors.

There are 100 numbers from 801 to 900. If you list the amount of factors for each number, then arrange those amounts from smallest to largest, the amounts that will appear in the 50th and 51st spots will both be 6. That means that 6 is the median amount of factors. If we had different amounts in the 50th and 51st spots, we would average the two amounts together to get the median.

If you add up the amounts of factors that the numbers from 801 to 900 have, you will get 794. If you divide 794 by 100, the number of entries, then you will know that 7.94 is the mean amount of factors.

What about the mode? Which amount of factors appears the most? That’s why we are having a Horse Race, to see if more numbers have 2 factors, 3 factors, 4 factors, or a different amount of factors. So pick your pony. We’ll see which amount wins, and we’ll find out what the mode is at the same time.

The contenders are these amounts: 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 27, 32.

I should tell you that only perfect squares can have an odd amount of factors, so you probably don’t want to pick an odd amount.

Here are some interesting facts about the numbers from 801 to 900 that might help you decide which pony to pick.

  • We had the smallest two consecutive numbers with exactly 12 factors: (819, 820)
  • We had the fourth prime decade: (821, 823, 827, 829). All four of those numbers are prime numbers and have exactly two factors.
  • We had five consecutive numbers whose square roots can be reduced: (844, 845, 846, 847, 848). Three of those numbers had 6 factors, one had 10, and one had 12.
  • We also had 840, the smallest number with exactly 32 factors
  • 900 is the smallest number with exactly 27 factors. Coincidentally, the amount that is the mode will appear 27 times.

As the following table shows, there are 42 integers from 801 to 900 that have square roots that can be simplified. 42 is more than any previous set of 100 numbers has given us. Even still we are still holding close to just under 40% of integers having square roots that can be simplified.

Okay. If you’ve picked your pony, NOW you can watch the Horse Race:

900 Horse Race
make science GIFs like this at MakeaGif
Hmm…

The race was exciting for a second or two.

As you can see from the Horse Race the mode is 4. How did your pony do?

Here’s a little more about the number 900:

900 is the sum of the fourteen prime numbers from 37 to 97.

24² + 18² = 900

900 is the hypotenuse of two Pythagorean triples:

  • 252-864-900, which is 24² – 18², 2(24)(18), 24² + 18². It is also (7-24-25) times 36.
  • 540-720-900, which is (3-4-5) times 180.

900 is the sum of the interior angles of a heptagon (seven-sided polygon).

  • 900 is a composite number and a perfect square.
  • Prime factorization: 900 = 2 × 2 × 3 × 3 × 5 × 5, which can be written 900 = 2² × 3² × 5²
  • The exponents in the prime factorization are 2, 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 3 = 27. Therefore 900 has exactly 27 factors.
  • Factors of 900: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900
  • Factor pairs: 900 = 1 × 900, 2 × 450, 3 × 300, 4 × 225, 5 × 180, 6 × 150, 9 × 100, 10 × 90, 12 × 75, 15 × 60, 18 × 50, 20 × 45, 25 × 36, or 30 × 30
  • Taking the factor pair with the largest square number factor, we get √900 = (√30)(√30) = 30.

 

899 and Level 3

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Yesterday I worked with a student who knew that 5 × 5 = 25 but couldn’t remember what 4 × 6 is. I didn’t use the variable “n”, but I used some examples from the times table to help her understand that (n-1)(n+1) = n² – 1. Always. Then I said to her, “30 × 30 = 900, so how much is 29 × 31?” She answered, “899?” I told her, “Yes, it is.” She was pretty pleased with herself.

29 and 31 are twin primes, so that makes 29 × 31 = 899 even cooler.

899 is also the hypotenuse of a Pythagorean triple:

  • 620-651-899 which is (20-21-29) times 31.

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 899 is a composite number.
  • Prime factorization: 899 = 29 × 31
  • 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 899 has exactly 4 factors.
  • Factors of 899: 1, 29, 31, 899
  • Factor pairs: 899 = 1 × 899 or 29 × 31
  • 899 has no square factors that allow its square root to be simplified. √899 ≈ 29.9833

898 and Level 2

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27² + 13² = 898. That means that 898 is the hypotenuse of a Pythagorean triple:

  • 560-702-898, which is 2 times (280-351-449).

898 reads the same way frontwards and backwards so it is a palindrome in base 10.

It is also palindrome 747 in BASE 11 because 7(11²) + 4(11) + 7(1) = 898

AND it is palindrome 1G1 in BASE 23 (G is 16 in base 10) because 1(23²) + 16(23) +1(1) =898.

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 898 is a composite number.
  • Prime factorization: 898 = 2 × 449
  • 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 898 has exactly 4 factors.
  • Factors of 898: 1, 2, 449, 898
  • Factor pairs: 898 = 1 × 898 or 2 × 449
  • 898 has no square factors that allow its square root to be simplified. √898 ≈ 29.9666

 

897 and Level 1

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(3 + 0)(3 + 10)(3 + 20) = 897

897 is the hypotenuse of a Pythagorean triple:

  • 345-828-897, which is (5-12-13) times 69.

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 897 is a composite number.
  • Prime factorization: 897 = 3 × 13 × 23
  • 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 897 has exactly 8 factors.
  • Factors of 897: 1, 3, 13, 23, 39, 69, 299, 897
  • Factor pairs: 897 = 1 × 897, 3 × 299, 13 × 69, or 23 × 39
  • 897 has no square factors that allow its square root to be simplified. √897 ≈ 29.949958.

896÷8=112. Math Teachers, It’s Carnival Time!

/ ivasallay / 10 Comments
  1. Welcome to the 112th Math Education Blog Carnival! There are many fabulous rides at this carnival, and hopefully you won’t get motion sickness on any of them!

At this first booth, we have the mystifying number 112. What is amazing about the number 112?

Well, not only is 112² = 2² + 4² + 6² + 7² + 8² + 9² + 11² + 15² + 16² + 17² + 18² + 19² + 24² + 25² + 27² + 29² + 33² + 35² + 37² + 42² + 50², but 112 is also the side length of the SMALLEST square that is composed entirely of smaller distinct sized squares with integer sides. Sources: Squaring a Square and OEIS.org. There are 21 different squares in this square with side length 112. Click on the image below and it will magically become bigger. You can then print it, cut it into pieces to make a puzzle, and take that home as your first carnival prize today.

I don’t know if it is significant, but 112 is 7 × 4², and 112’s square was divided into 3 × 7 different squares . 175 is also such a side length, and 175 is 7 × 5², and 175’s square was divided into 3 × 8 different squares.

 

Addition, Subtraction, Multiplication, Division

Bedtime Math has a quick math activity for kids every evening. Here’s one about adding, subtracting, and/or multiplying at the Giraffe Hotel.

Algorithms

Rodi Steinig leads a math circle of students ages 11-17 in a course titled “Our Algorithmic Culture,”. It may surprise you that algorithms are not just for math; they are for real, real life, too. See for yourself as several activities are described in Introducing Algorithms, a post about the first of 8 sessions on the subject.

Math Art

RobertLovesPi’s blog regularly features a rotating solid geometric shape or a beautiful tessellation such as this one that can be enjoyed by young or old alike.

David Mitchell of Latticelabyrinths explains how he and his friend, Jacob, made a beautiful structure for the September 1917 Wirksworth Art and Architecture Trail using a large peg board, pegs and 1302 red or blue precisely-cut wooden equilateral triangles. Amazing! I wish I could have seen it in person.

What is a Rotogon? Katie Steckles of aperiodical.com blogged about this beautiful computer generated, constantly transforming piece of art that could mesmerize young and old alike.

Bell Ringers

What is the Same? What is Different?  has a wide variety of thoughtful activities that can get your students’ brain juices flowing.

Try Math Visuals for other great bell ringers.

Life Through a Mathematician’s Eyes sees a great deal of beauty in Pascal’s Triangle, but it isn’t the significant part of the curriculum she wishes it could be. She got around that though. In her post, Pascal’s Triangle, she shared some great articles that she turned into bell ringers for her students to contemplate when they arrive to class. She also made assignments that can be completed in class or at home.

Bulletin Boards

Paul Murray creates bulletin boards near the lunch room with questions on them.  Some of the questions are math problems. For the last ten years or so, students have stopped by his bulletin boards, read them, and pondered the questions. Read how he does it in A Math Bulletin Board that Actually Gets Read!

Carnival of Mathematics

Earlier this month Just Maths published the more advanced 149th Carnival of Mathematics. It has several great links in it that could pique high schoolers’ interest even if those students aren’t able to understand all the mathematics yet. Next month Alexander at CoDiMa will host the 150th Carnival of Mathematics.

Math Competitions

Some people like entering math competitions. If you have a student that likes them, look at this post from Resourceaholic. It has questions that can be a fun challenge whether you like competitions or not.

Games

Alan Parr often plays his envelop game with students who are learning many different mathematical concepts. His students all enjoy it. The game described in A Wow! Conversation with Amy was easy to put together, let Amy display some brilliant reasoning and provided its creator a very memorable experience, perhaps his most memorable this school year. .

Geometry

Paula Beardell Krieg of Bookzoompa’s wrote a post about Symmetry for 4 – 5 year olds that I adore. Even kids that young can make some gorgeous geometric art.

Mike’s Math Page does so many great math videos with his sons that its difficult to pick only one or two. I went with a couple of posts about geometry: Playing with Some Mathy Art ideas this morning which will appeal to kids of all ages and Lessons from a great geometry homework problem for older kids.

Before teaching congruent triangle proofs, Mrs. E Teaches Math recommends a paper folding, cutting, and arranging activity that helps students visualize overlapping triangles.

Mathematical Humor: 

A little mathematical humor can help students access prior knowledge, or it can make a new concept memorable. Joseph Nebus reads lots of comics to find ones with some mathematics in them. This August 17, 2017 edition of Reading the Comics has a few that could help students remember what irrational means, what sum means, or what the < sign means.

Math Literature Books

Math Book Magic introduces us to a new book, Stack the Cats by Susie Ghahremani. The book is just right for kindergarten and earlier elementary grades. It can help you talk to kids about counting, adding, even decomposing numbers in a very fun way. Kids may not even be aware they are learning math as they find different ways to stack toy cats and other objects. Stack the Cats joins the #Mathbookmagic

Ben Orlin of Math with Bad Drawings delights us with Literature’s Greatest Opening Lines, as Written By Mathematicians.

Denise Gaskins has written Word Problems from Literature. It has a more serious but still very fun approach to exploring mathematics and problem solving through good literature.

Math Magic

Alex Bellos of Science Book a Day introduces us to Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks. Every trick in the book introduces a different mathematical idea, and lots of magician secrets are revealed. Sounds fun!

Managing the Mathematics Classroom

Mrs. G. of Give Me a Sine blogged that this school year she had the best first week of school she’s ever had. She mixed some of her tried and true favorite activities with some new ones that she learned this summer from NYC Math Lab and Sara Vanderwerf. Mrs. G gives specifics detailing what she did. Her students’ response described in the next to the last paragraph is enviable.

Years ago I made a seating chart for a Pre-Algebra class. I didn’t know the students yet and arranged my seating chart in alphabetical order by first name. It was a disaster. I had placed the tallest student in the class at a desk in front of the shortest student in the class. Once one legitimate complaint was voiced, other complaints followed. If I could have read Mrs. E Teaches Math post How to Create a Seating Chart before that school year, I would have had one less frustration.

Math Music

Coleen Young has updated the Mathematical Songs on her website to include MinuteMath’s quadratic formula sung to a One Direction song. She wrote that the song makes her students smile and sing along.

Number Sense

What’s your favorite number? John Golden of Math Hombre asked that question to people and got some wonderful responses. Really cool mathematics is attached to several numbers including the Top Ten Favorite Numbers.

Problem Solving

True problem solving is more about “Why?” than it is about an exact “What?” Denise Gaskins of Let’s Play Math has written several questions to teach students How to Succeed in Math: Answer-Getting vs. Problem-Solving

Puzzles

Long ago mathematicians were often philosophers and philosophers were often mathematicians. Simona Prilogan is both. Every day she posts a mathematical meme, a puzzle such as this one, and a philosophical mathematical thought. Search her site for all kinds of goodies to appeal mostly to middle school children and up.

Alok Goyal’s Puzzles Page shared a puzzle titled 10 Friends. Upper elementary students will be able to understand what the puzzle is asking but would probably need a lot of guidance to solve it.

Rupesh Gesota of Math Coach shared an interesting 6 rectangle puzzle and revealed several different methods students used to solve the puzzle.

My blog typically features a factoring puzzle such as this one that I fancied up for back to school:

This month I did something different: I wrote an elementary-school-age time management lesson plan with an object lesson that uses Tangram puzzle pieces. It was a big hit with the teachers and the students.

Resources

Three J’s Learning wrote math recommendations for 3 year-olds! It includes a list of measuring devices a 3 year-old would love to use and learn from.

Singapore Maths Tuition has assembled a list of the Best Online Resources to Improve Your Math Skills along with their pros and cons.

Resource Room Dot Net Blog wrote about an experience using Illustrative Mathematics (a resource for 6-8 grades) and how it is worth the effort to give feedback.

Square Roots

Last month’s 111 Math Teachers at Play Carnival was hosted at High Heels and No. 2 Pencils by Jacqueline Richardson. This month the Amazing Jacqueline doesn’t just guess your age, she can guess the square root of your age within a few decimal places! I have never seen anything like this before. She uses tiles and grid paper to model square roots of non-perfect squares and is amazingly accurate. Teach your students this way, and they will be amazed. They will understand square roots so well that they will be amazing as well.

Teaching Practices

What are the zeros of this polynomial? Julie Morgan of Fraction Fanatic lets her students give their solutions in a colorful way. This practice has become her new favorite.  Lots of discussion happens. Young children could also use this fun method to give answers.

Ed Southall of Solvemymaths wrote a sobering post titled Sorry we keep lying to you… about lies our math teachers told us and we continue to perpetuate. Read it. Share it. Maybe the lies will stop.

Sara Vanderwerf explains how her Stand and Talks engage students much more than Think/Pair/Share does and REALLY gets the whole class talking about math and contributing to a whole class discussion.

Statitistics

Business blogger Lanisha Butterfield wrote a fascinating article titled Arithmophobia. A major portion of the article was an interview with statistician Jennifer Rogers who did well in math as a kid but HATED it until she was introduced to A-level maths and statistics in school. High School teachers and students should be especially interested in this article.

Sue VanHattum of Math Mama Writes detailed her first day teaching algebra, statistics, and calculus this year. When she discussed the class syllabus, she inserted some fun mathematics here and there. She summed up that first day and shared every math teacher’s universal dream, “I think the class went well. If they really feel good about it, they’ll end up thinking I’m their best teacher ever.”

To everyone who plays at this Carnival, I hope your students think that way about you!

Thank you to everyone who blogged about teaching mathematics this month, and especially thank you to those who submitted a post to this carnival.

I’d like to encourage everyone who blogs about math to submit a post to next month’s carnival which will be hosted at Three J’s Learning.

***************

Since this is my 896th post, I’ll also write a little bit about that number which happens to be 8 times 112. A factor tree for 112 is contained in this factor tree for 896.

896 is SS in BASE 31 (S is 28 base 10), because 28(31) + 28(1) = 28(31 + 1) = 28(32) = 896.

896 is also S0 in BASE 32 because 28(32) + 0(1) = 896.

896 is the sum of six consecutive prime numbers: 137 + 139 + 149 + 151 + 157 + 163 = 896.

Here is the factoring information for 896 with the ten factors of 112 in red.

  • 896 is a composite number.
  • Prime factorization: 896 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7, which can be written 896 = 2⁽⁴⁺³⁾ × 7 or 896 = 2⁷ × 7
  • The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 × 2 = 16. Therefore 896 has exactly 16 factors.
  • Factors of 896: 1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 128, 224, 448, 896
  • Factor pairs: 896 = 1 × 896, 2 × 448, 4 × 224, 7 × 128, 8 × 112, 14 × 64, 16 × 56, or 28 × 32
  • Taking the factor pair with the largest square number factor, we get √896 = (√64)(√14) = 8√14 ≈ 29.933259

 

895 and Level 6

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895 is the hypotenuse of Pythagorean triple 537-716-895, which is (3-4-5) times 179.

895 is also palindrome 292 in BASE 19 because 2(19²) + 9(19) + 2(1) = 895.

Print the puzzles or type the solution on this excel file: 12 factors 886-896

  • 895 is a composite number.
  • Prime factorization: 895 = 5 × 179
  • 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 895 has exactly 4 factors.
  • Factors of 895: 1, 5, 179, 895
  • Factor pairs: 895 = 1 × 895 or 5 × 179
  • 895 has no square factors that allow its square root to be simplified. √895 ≈ 29.91655

894 and Level 5

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894 is the hypotenuse of a Pythagorean triple: 306-840-894

894 is also palindrome 626 in BASE 12 because 6(144) + 2(12) + 6(1) = 894

Print the puzzles or type the solution on this excel file: 12 factors 886-896

  • 894 is a composite number.
  • Prime factorization: 894 = 2 × 3 × 149
  • 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 894 has exactly 8 factors.
  • Factors of 894: 1, 2, 3, 6, 149, 298, 447, 894
  • Factor pairs: 894 = 1 × 894, 2 × 447, 3 × 298, or 6 × 149
  • 894 has no square factors that allow its square root to be simplified. √894 ≈ 29.89983

A little surprise is waiting when you square 893.

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What is special about the number 893? It makes a pretty cool square with two 4’s, 7’s, and 9’s in it. Thank you OEIS.org for that fun fact.

  • 893 is a composite number.
  • Prime factorization: 893 = 19 × 47
  • 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 893 has exactly 4 factors.
  • Factors of 893: 1, 19, 47, 893
  • Factor pairs: 893 = 1 × 893 or 19 × 47
  • 893 has no square factors that allow its square root to be simplified. √893 ≈ 29.8831