On Wednesday I read an article in my local paper titled From undrafted free agent to the biggest stage: Britain Covey’s ‘unique ride’ to Super Bowl LVII. It got me quite excited for today’s game. Unfortunately, between Wednesday and Sunday, Covey suffered a hamstring injury. Getting a Super Bowl Ring certainly isn’t easy. Being able to contribute to your team’s winning the game isn’t easy either. UPDATE: Good News! He was able to make at least one play in the second quarter! Further update: Even though his team lost the game in the final minutes, this rookie played well when he was on the field. I think both teams played exceptionally well and gave us all an exciting game to watch.
Before the game, it is a mystery which team will win the game. The difficulty level of this puzzle is a mystery also.
If you look at today’s puzzle just right, I think it looks a little like a super bowl ring, but I forwarn you, it will not be easy to get this puzzle either. You will need to place the numbers from 1 to 12 both in the first column and in the top row so that the given clues are the products of the numbers you place. Logic and practice will get you there. Good Luck!
You also might enjoy this next Super Bowl puzzle I saw on Twitter:
1747 has no exponents greater than 1 in its prime factorization, so √1747 cannot be simplified.
The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1747 has exactly 2 factors.
The factors of 1747 are outlined with their factor pair partners in the graphic below.
How do we know that 1747 is a prime number? If 1747 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1747. Since 1747 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1747 is a prime number.
More About the Number 1747:
1747 is a palindrome in base 15.
It’s 7B7₁₅ because 7(15²) + 11(15) + 7(1) = 1747.
The last 10 seconds of the year, we like to countdown from 10 to the new year. I like a mathematical way of counting down so I try to make an equation with the numbers from 10 to 1 that equals the coming year. This year I could have based my countdown on last year’s countdown and said
(10-9+8×7×6)(5-4)(3)(2)+1 = 2023,
but this is a blog about factoring so I want a countdown that takes you to the prime factors of 2023 first. Here’s my countdown: (Note: Even though I used 1 as a factor twice in the countdown, I am very much aware that 1 is not a prime factor of any number.)
2023 has something in common with 2022. When either number and their reverses are squared, something interesting happens…it’s almost like looking in a mirror!
Only 50 numbers less than 10000 can make a similar claim to fame:
Factors of 2023:
2023 is a composite number.
Prime factorization: 2023 = 7 × 17 × 17, which can be written 2023 = 7 × 17².
2023 has at least one exponent greater than 1 in its prime factorization so √2023 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √2023 = (√289)(√7) = 17√7.
The exponents in the prime factorization are 1 and 2. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1) = 2 × 3 = 6. Therefore 2023 has exactly 6 factors.
The factors of 2023 are outlined with their factor pair partners in the graphic below.
More About the Number 2023:
What do 2023 tiny squares look like?
2023 is the sum of consecutive numbers in five different ways:
And it is the sum of consecutive odd numbers in two ways:
2023 is a palindrome in base 16 because
7(16²) + 14(16) + 7(1) = 2023.
This tweet demonstrates that the prime factors of 2023 have a relationship with the digits of 2023.
2023 is the only number that is equal to the sum of its digits multiplied by the square of the sum of the squares of its digits, where the sum of its digits and the sum of the squares of its digits are also its prime factors.#HappyNewYear#HappyNewYear2023pic.twitter.com/fHnd6nEQhp
That might seem like a lot of mathematical mumble jumble, but with a little bit of explanation, it can be understood. And even though I made the problem look scarier because I substituted 2+0+2+3 for 7, some older elementary students who already understand powers, factorials, and/or remainders, will get it. I’m confident you can too.
You could also give the following list of facts to older elementary students and ask them to use it to find the remainder when they divide 823,543 by 5040.
Because its factors, 17 and 289, are hypotenuses of Pythagorean triples, 2023 is also the hypotenuse of some Pythagorean triples:
952-1785-2023 which is 119(8-15-17) , and
1127-1680-2023 which is 7(161-240-289).
Ureczky József also pointed out in the comments of this post, that 2023 is the short leg in SIX Pythagorean triples, and thus
2023² = 2046265² – 2046264²
2023²= 292327² – 292320²
2023² = 120377² – 120360²
2023² = 17255² – 17136²
2023² = 41785² – 41736²
2023² = 7225² – 6936²
One of those triples is a primitive triple. Can you determine which one?
Ureczky József shared one more amazing fact in the comments that I’m replicating here:
More Mathematical Tweets About 2023:
These tweets are more or less in the order I saw them, not in order of mathematical difficulty. I will add more as I see them.
2023 is coming up so here’s a fun math fact about that number:
It only has one hole in it so it’s going to look ridiculous if you make New Year’s glasses out of it.
For calculus teachers looking for a math-dorky integral problem for when school restarts in January, revise this problem to find the location of the vertical line for the function y = x^2022. The two areas will be equal at x=2022/2023 !
2023 is the sum of the reverse of 3 consecutive primes (931+941+151), has consecutive pairs of digits that each sum to a prime (2+0, 0+2, 2+3), and prime factors that concatenate to make a palindrome (2023 = 7×17×17 → 7||17||17 = 71717).#HappyNewYear#HappyNewYear2023
A nice thing about 2023 is that it may look like a prime number, but it is not. A nicer thing is that 4 years later we get a prime number for a whole year. And as a bonus it is a twin prime, so we just have to endure 2028 to reach its twin 2029
The number 2023 is 17*7*17
17 is special because it’s the only prime number which is the sum of four consecutive primes. You probably know why 7 is.
There are many other reasons why these numbers are special.
Thank you. You’re welcome.
Correction: Thanks to @SirmaRoca mentioning there’re more, I did find three more using algebra and then I used wolfram to verify:
(9 solutions in total)
The number 2023 may seem rather undistinguished, but its prime decomposition is interesting. It contains three (lucky) 7s and two 1s: 7*17*17, which of course portends ……….. nothing. In any case, Happy 2023!
I started this a few days ago and it’s going pretty well! Tried to keep the numbers 2023 in order for as long as I could. If anyone has found solutions for 53, 78, 83, 85, 89, 91, 92, 93, and 95, I’d love to see them. I’m currently stuck on those! https://t.co/GnlNDdcrYXpic.twitter.com/arOwkxgmZT
Happy New Year, #MathClub! If you’re like us, you’re probably wondering if 2023 is prime. Let’s check if it’s divisible by seven (“dbs”). 2023 is dbs iff 202 – (3 + 3) = 196 is. In turn, 196 is dbs iff 19 – (6 + 6) = 7 is. So 2023 is not prime, but 7 is a pretty lucky factor!
A test for ÷ by 7:
Split number in groups of 2 digits from the right, 1st group is x1, 2nd x2, 3rd x4 etc. 2023 becomes 20×2+23×1 = 63 & since 63 =9×7, 2023 is ÷7 too.
Note: this is based on 100/7 = 14 R 2 so 2 = multiplier base & 100 is 10^2 giving groups of 2
Chasing Unicorns humorously blogs about Organizing Jelly Beans. How many jelly beans can you eat each day to keep yourself below the estimate of refined sugar consumed per American per day?
Within 1%, how long is the hypotenuse of this right triangle? If certain criteria are met, John D. Cook’s blog post, Hypotenuse Approximation, can help you be the first to find the correct answer and win the prize.
Fractions, Ratios, and Decimals
Henri Picciotto of Henri’s Math Education Blog updates us on how to use fraction rectangles to help students make sense of adding, subtracting, or comparing fractions with different denominators.
1001 Math Problems shares an engaging and delicious Chocolate Problem involving fractions.
Read how much laughter can be had learning long division involving decimals in FiveHundredaDay’s post It’s not them, it’s me.
Ajitadeshmukh shares the game, The Number Detective [Spying the number]. This is a game that uses an ordinary deck of playing cards and reinforces the concepts of adding, subtracting, multiplying, and/or dividing. It can be played by children in early elementary grades and up.
Primary Ideas shared how well a game of Noggle (Number Boggle) went when it was played in a Google session remotely.
Anna of one+epsilon designed a logic game called Dot, Dot Poof! Here’s a bonus: Kids 6 and up might inadvertently learn a little linear algebra playing it, too!
Wendi Bernau made an Easter Egg Hunt Escape Room for her 15- and 17-year-old kids. The escape room included puzzles based on their current schoolwork. The 17-year-old had to solve a puzzle that required calculations, graphing, and trigonometry. The kids liked the escape room so much that they are already talking about doing it again next year.
Indrajit RoyChoudhury tells us about Bhaskaracharya, a 12th century Indian mathematician and astronomer in Arjuna’s Arrows and Algebra. Bhaskaracharya discovered differential calculus 500 years before the births of either Newton or Leibniz.
LA of Waking up on the Wrong Side of 50 is featured in the controversial current events area of the Math Museum in Anything Can Happen Friday: Math. LA includes the actual newsletter in which Oregon instructs its math teachers to allow for more than one correct answer. LA is upset thinking that now Oregon math teachers must accept incorrect math like 2 + 2 = 79. Perhaps Oregon is just welcoming some of Denise Gaskins’ math rebels who might say that 2 + 2 = 79 – 75, or some other of the infinite number of possible non-simplified yet still very much correct answers.
Likewise, the College Fix reported that Oregon math teachers have been instructed to let their students show their work by making TikToks, silent videos, or cartoons about the math they are learning, in other words, let students make their own math. I think about Ramanujan who taught himself math from an old textbook and then created his own mathematical symbols and terminology when he dreamed up more advanced mathematics. Later when he was told he needed to prove his ingenious mathematical formulas with rigorous proofs, did it help him or restrict him?
told my students today we could film a tic tok today after we did our math lesson next week. I’ve never seen kids so excited to do math 😭😭😭
“What an astonishing thing a book is. It’s a flat object made from a tree with flexible parts on which are imprinted lots of funny dark squiggles. But one glance at it and you’re inside the mind of another person, maybe somebody dead for thousands of years.” – Carl Sagan ♥️ pic.twitter.com/dX9EEsWftx
1 + 1 + 1 = 3. Any number greater than one can be partitioned in a similar fashion. Patricia Nozell reviews a perfect picture book, I Am One: A Book of Action by Susan Verde. A little math can be learned while one person works with another and another to make the world a better place.
Writing this post has introduced me to Perfect Picture Book Fridays. Susanna Leonard Hill reviewed Little Ewe: The Story of One Lost Sheep, by Laura Sassi. Your 3- to 5-year-old will love counting logs, frogs, and other rhyming nouns as you read this book together.
Sue Heavenrich of Sally’s Bookshelf blogged about Bracelets for Bina’s Brothers, a picture book about estimation for 3-6-year-olds, and concluded that Math + Art > Numbers. Activities to make the math in the book more meaningful are also included in the blog post.
Patricia Tilton of Children’s Books Heal reviewed Wonder Women of Science by Tiera Fletcher and Ginger Rue as part of Women’s History Month. The book is perfect for 9 to 12-year-olds. She also made me aware that Nerdy Book Club reviewed the same book. From that review, I learned the delightful true story of a human calculator named Tiera Fletcher that I am anxious for you to read as well!
In Monday’s Math Madness, Willow Croft thoroughly enjoys a 15th-century maritime manuscript called The Book of Michael of Rhodes. There is a lot of math in the book, but even if the reader doesn’t like math much, it won’t take away from the thrilling adventure. It is suitable for high school students and older.
Kelly Darke of MathBookMagic and FairyMathMother would like you to know about Math Book Wisdom: An Early Math Resource Book. It isn’t a book to read to kids, but it is filled with math wisdom for the parents and teachers who teach children.
Today’s Gr 6 chat q: “tell us an amazing fact.” One S shared about the world’s current oldest living person, Kane Tanaka. from wiki:
“Kane said she would like to live to the age of 120, crediting family, sleep, hope, eating good food & practicing mathematics for her longevity.”
When a student didn’t understand a mathematical concept, he broke a rule by leaving the classroom. Kaneka Turner of BlackWomenRockMath details how she listened to the student with her ears, her eyes, and her heart in The Art of Listening. By so doing, she successfully helped him make the connections needed to understand the lesson while simultaneously letting him know he was truly understood. What trajectory would his life be on now, if she had not listened as she did?
In the second half of Bill Davidson’s podcast interview with Robin Ramos, she describes how she scripts questions and listens to not just individual students but to a classroom of students at the same time!
My heart broke when I read Looking at Love Lost, by murisopsis of A Different Perspective. It is a poem about falling out of love with mathematics in high school beginning with trigonometry. Simply saying Trig is Easy doesn’t help and only makes a person not feel heard. Perhaps Wyrd Smythe’s Explanation of Trig Basics might have been helpful?
Sarah Carter of MathEqualsLove shares a new puzzle in Number Ball Puzzles by Naoki Inaba. She translates the rules from Japanese to English so that you can have some idea where to put the missing numbers in the puzzle. Be warned, for the bigger puzzles, you might need to use your eraser a lot.
Sara also shared a sequence puzzle. Her students have enjoyed predicting the next letter in the sequence.
Did you know that if you get 11,000 steps a day, you will walk a million steps every quarter and just over 4 million steps a year? LisaFeatherstone had a daily goal of walking 10,000 steps and still made the 4-million steps goal. She used a spreadsheet to track the data her fitbit gave her and wrote a formula to predict when she would meet her goal.
Lvonlanken of The Shy Genealogist analyses the data she’s collected to determine which John Smith is her ancestor in Sorting the Land Records. Some genealogical programs will provide you with all kinds of statistics from your family tree. See the stats the Chiddicks Family found in My Family Tree in Numbers. I was pleased that they didn’t simply accept every statistic. They made predictions of the results and compared their predictions with the statistics the program produced.
Jo Morgan of Resourceaholic recently celebrated seven years of blogging by reviewing the very best teaching ideas and resources from the previous year and naming the winners of her (Maths) Gem Awards. Check it out!
A short story, Advanced Word Problems in Portal Math, is a finalist in the Nebula Best Short Story Contest. The reviewer didn’t care for the story because the math references were hard to understand. Let me tell you a little secret: I think that’s the way it was meant to be because I didn’t get the math references either! The story was just a fun way to make math. Another example of purposeful over-our-heads math was in a Barnaby comic. I know how to find the determinant of a two by two matrix and how to multiply binomials, but I look forward to Joseph Nebus explaining that comic sometime soon. It is still a funny comic even if I don’t fully understand it yet.
I hope you had a wonderful time at this month’s carnival! This month the Carnival of Mathematics #192 was hosted at Eddie’s Math & Calculator Blog. Perhaps you would like to design your own carnival.
I know this Design-A-Playground Jamboard will be a hit because it was incredibly fun to make. Students can design their own or remix my playground. Anyone want to design a playground for Ss to remix on this Jam? If so, post “I want to Jam with you,” and I’ll DM you a link. pic.twitter.com/En36RnKiDr
Simran M Karkera of MSCNM tells the story of a girl who loved math that used trigonometry and calculus to design a roller coaster that thrilled her previously-mocking friends in A Mathematical Ride!
Last month the 145th Playful Math Carnival was hosted by Mathhombre. Perhaps you would like to host the next carnival or one later in the year. You don’t have to go overboard like I probably did. I was having so much fun, I couldn’t stop myself! To volunteer to host a carnival go to Denise Gaskins’ Carnival Volunteer Page.
Tangram Tuesday! Have you heard of the magic egg? It‘s a shape puzzle with 9 pieces. We‘ve had a go at it today in school and the verdict is it‘s tougher to make an egg than a tangram square! What do you think? Have a look on nrich for a copy of the egg. @ReayPrimary pic.twitter.com/vg5Io2jeRm
I luv that my student googled egg puns to name these!
Just thought you’d appreciate knowing this was a hit with my Ss as our opening Desmos screen today! We are F2F so there was a lot of chatter around the room with their creative names – so fun! pic.twitter.com/zDYzXdHNQA
Thought I was done with decorating #EasterEggs2021, but hadn’t thought about #MathsArtMonday.Since I had eggs on my mind, made these today. I like how the one with curve stitching looks like it’s wrapping around the egg like a sweater. The other is based on convolute of a circle pic.twitter.com/AY0LepOAYJ
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.
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.
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?
I’ve taught Algebra students about Systems of Linear Equations for years. I’m also a big fan of paper folding.
I recently had an “ah-ha moment” when I realized that these two things are closely connected.
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?
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.
Here are some small puzzles based around divisibility, with arrows revealing which number divides which. Dots indicate that neither divides the other.
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.
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.
Excellent, informed @tes piece from @NumCog
“when children tend to hit Year 1, they are usually expected to be able to use numerical symbols for basic arithmetic when, for some children, they might not be at that point yet.” >> https://t.co/FwMIupazHb
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?
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.
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?
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.
How beautiful circles in love can be
Holding together hand in hand
Hiding the triangles their good old friends
A face like a cathedral stained glass
Yellow and green, like jewels from a palace.
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?
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.
In A Neat Unsolved Problem in Number Theory That Kids Can Explore, Mike’s Math Page explores the new-to-me Collatz conjecture that for every positive n, the sum 3 + 8n will equal a perfect square plus an even number. It’s a simple enough conjecture for kids to understand and it is fascinating, yet mathematicians have not been able to prove or disprove it yet!
Subha laxmi Moharana (Angel Subu) writes creatively about some tough topics in high school mathematics in Math Poem. I think her words could be turned into a rap.
Joyful Parenting made a simple kindergarten-age counting game and called it Snack Math, but even older kids might enjoy figuring out exactly how many crackers are required to play the game.
How many are in the jar. What is a good estimate? Add Steve Wyborney’s clues one by one to get an even better estimate. He has 51 New Esti-Mysteries that also happen to teach several different math concepts.
House of Mirrors (Reflecting on Mathematics Teaching)
We often reflect on the effectiveness of our teaching methods. Sometimes we are advised to require students to use more strategies. We might ask them to notice or wonder about a concept. These two thoughtful posts will certainly give you cause for reflection:
What is your favorite part of a cupcake? What if you could buy just that part? What if you wanted to put a whole cupcake together? How much would that cost? Your child can learn about money and decimals exploring those answers with Mathgeekmama’s Money Math Problems.
Puzzle: The Largest Value in Coins Without Change for $1
Beads can be a fun manipulative when learning mathematics. Joseph Nebus has begun his 2019 Mathematics A-Z series by writing about the Japanese abacus. He compares it to a slide rule and the Chinese abacus. He also describes how to use it to add, subtract, and multiply numbers. Students could have some fun using it to understand place value, too.
Life Through a Mathematician’s Eyes is giving museum tours in A History of Mathematics-August. K-12 students could be fascinated by the mathematical relics from the Smithsonian founded in August 1846 as well as the Seven Bridges of Königsberg solved by Euler in August 1735.
Alan Parr writes about a newspaper puzzle called Evens Puzzles. He suggests that students can make their own and hints that he has thought up several variations of it. I look forward to reading about those!
American Calendars for September had more than a week’s worth of palindromes. Would palindromes make a good puzzle? Yes! Print off a 100 chart and try Denise Gaskins’s A Puzzle for Palindromes. Also, check out her new Morning Coffee feature each week for more math teaching tips.
Next Month’s Carnival
That’s it for this month’s Math Education Blog Carnival. The 132nd Carnival will be next month at Arithmophobia No More. Would you like to share a post or host the carnival? Go to Let’s Play Math for details!
Ladies and gentlemen welcome to the Playful Math Education Blog Carnival featuring the incredible number 127 of the famous Mersenne Prime family! Let’s give a big hand to. . . . .
2²-1 = 3, the smallest single Mersenne Prime;
2³-1 = 7, the smallest double Mersenne Prime;
2⁷-1 = 127, the smallest triple Mersenne Prime; and finally
39-digit 2¹²⁷-1, the smallest quadruple Mersenne Prime!
It took NINETEEN YEARS (1857 to 1876) for Édouard Lucas to test Mersenne Prime Number 2¹²⁷-1 BY HAND to successfully verify that it is indeed a prime number. It is the largest Mersenne Prime that has ever been verified by hand calculations!
Ladies and gentlemen, Prime number 127 has one more amazing feat up its sleeve that you will have to see to believe:
In this month’s blog carnival posts, the amazing Desmos will delight and astound young and old alike. There will even be an Easter egg hunt! The blog post links (in turquoise) are joined by several links from Twitter (in blue-violet) and a fewfrom other places such as Youtube (in red). Stay as long as you like and ENJOY what the many carnival participates have to offer in 20 different amusement areas!
After this year’s very long winter, I’m especially glad Colleen Young collected some lovely and amazing springtime Desmos drawings and gifs in It’s Springtime. . . .
Do you see mathematics everywhere? Continuous-Everywhere-but-Differentiable-Nowhere can and does, even on a student’s shirt. Read the story and see the t-shirt design replication in Desmos in Going Off the Beaten Path.
Wanting to have your students create an art project in Desmos? 1ntegration-by-Parts has given that assignment many times and has links for student directions to help them meet your expectations in Desmos Art Project (Update).
You must click on the turtle face link in Desmos Art! With just ONE equation, a magnificently detailed turtle face was produced in the Desmos calculator. I was so impressed. I tweaked that single equation by changing the number 16 to 7.29, added some color restrictions as well as equations to make a mouth and some eyes to produce my own Desmos Art piece I call Blue-eyed Beauty.
Many people have a personal story to tell that explains why they love mathematics. Through a Mathematician’s Eyes opens up and shares her experiences in My Story. What obstacles did she face? How did she feel about enjoying a subject so many others hated?
Philip Jose Pacis played with some math vocabulary and wrote a poem he titled Fractions about a fractured relationship.
Calendars and time are mathematical topics. How many other mathematical terms do you see in Maggie C.’s poem On Time? Do any of those words have more than one meaning?
Would you like to play Decimal Pickle? On Twitter, Mrs. Unger explained how to play it as well as a few of its variations.
Desmos First Aid Station?
There’s no need to call the paramedics when it’s time to learn about parametric functions. Suzanne Van Oy has come to the rescue! She sees a lot of value in parametric functions and their graphs. Why is she so excited about them? What’s all the fuss? Suzanne answers those questions and more in Why Parametrics?.
Suzanne Van Oy also recently put an incredible spinning Desmos Birthday Cake on twitter. How does she make Desmos gifs that don’t look like they need to be put on life support? Six months ago she blogged about how to do it in Making a Great Desmos Gif. She certainly knows what she’s doing!
Sometimes Desmos doesn’t do what you expect. Your work might need some first aid. DesmosGraph (Unofficial)’s post: Desmos Traps: Why Is It Not Working may have the diagnosis and cure you seek.
When teaching fraction division, should you start with rules or diagrams? Filling the Pail speaks from experience in Fraction Division and explains the advantages and disadvantages of both approaches.
Games and Educational Toys
On the spur of the moment, I came up with a very quick Yahtzee variation that I played with one of my students. We counted the number of rolls it took us to get a Yahtzee. Lowest score wins. He beat me badly every round as the graphic above this category attests, but on a different day, Lady Luck was with me more than him.
Denise Gaskin also has a tried and true Yahtzee game variation that she calls Six Hundred. You only need to print her directions and scoring sheets, provide six 6-sided die and pen or pencils, and you’ll be ready to make memories in more ways than one.
Autism Awareness Week was earlier in April. In this post, Special Educational Resources Blog reviewed three games made by Orchard Toys: Money Match Cafe (teaches about money), Look and Find Jigsaws (teaches number and letter recognition), and Bus Stop (uses processes like 3-2+4-1 to figure out how many kids are on the bus when the bus arrives at the bus station.)
The Mathematical Tourist shared how to play a game called Clobber. The game has been around since 2001, but the best strategy to win the game is still a mystery. I’m sure getting clobbered will be just as much fun as winning.
Wheeler’s Thoughts on Teaching used a bank balance problem to teach about solving a system of linear equations. The students were able to think about the problem and work on it with much fewer hints from their teacher. That makes teaching math much more fun for the students as well as the teacher.
Imagine this carnival ride: a catapult that will send you flying through the air! Lana Pavlova and Meredith Wilkes have assisted Math Book Magic in creating the perfect design of an unforgettable carnival ride in Play with Your Math with Little Pea. How far will this catapult take you?
Chirag Mittal took charge of April’s birthday celebration of Leonard Euler. Did you know that Euler is credited with being the first to use letters from our alphabet and the Greek’s alphabet to represent some very important functions and numbers: Σ, f(x), e, i, and π?
Alan Paar of Established 1962 has put together a tour of Wendover School and the way teachers taught and students there learned several mathematical topics and other subjects from 1868 to 1930.
Math with P. Nik gives instructions and several examples of his Three Elastic Bands puzzles. He said they were easy to make, so I made the one at the top of this category. Follow P. Nik’s instructions and you probably won’t need to click on the tiny answer key under the puzzle.
When Simona Prilogan of Fiat Lux writes a number puzzle, it is much more than it appears. You have to study patterns inside the puzzles to figure out what the relationship really is. Give her Wednesday Math Puzzle a try!
Simona included a bonus, information about Bolsover Castle, in her Monday Math Puzzle. You will find two different Math Puzzles in the middle of reading about the Castle!
This clever tie matching exercise from Math with P. Nik feels more like a puzzle than a worksheet. Can you match the graph families with the correct equation families?
Statistics and Probability
Yes, you can do statistics in Desmos! You can make Normal, Poisson and Binomial distributions and even graph box and whisker plots in Desmos! Colleen Young shows you what that looks like in her post Statistics with Desmos.
Does El Niño play much of a part in rising global temperatures? In New Kid in Town, Open Mind answers that question and includes line graphs to help us visualize global temperature data collected since 1979.
This year Easter occurred on April 21st. That seemed rather late to me, but it isn’t the lastest it could be. In Joseph Nebus’s post, What Dates Are Most Likely for Easter?, he’ll direct you to a post he wrote two years ago where all the data is lined up to figure out the probability.
What time is it? There is more than one valid way to give the correct time, and one way should not be labeled as a smarter way to give the time than the others. That’s the message given in Dan Meyer’s Don’t Teach Math “the Smart Way”. He even suggests a lovely game from Desmos to get kids talking about telling time.
After a long winter with snow causing several school days to be replaced with “e-learning days,” Educational Technology in Action wrote about using that same Desmos talking time activity in Desmos for meaningful e-learning days.
Thanks for coming to this month’s carnival! I hope you enjoyed it. I had a wonderful time hunting for goodies to put in the carnival and organizing it. I felt like I was on an Easter egg hunt looking for the best eggs!
Math Misery? will host May’s Playful Math Education Blog Carnival. Perhaps YOU will consider contacting Denise Gaskins and volunteering to host a future carnival! There are two open dates in the summer still available this year.
Does that pattern hold for all natural numbers? Could we claim that n² = n?
Yes, we can, and I’ve written a proof to prove it! The proof uses a valuable concept in mathematics called induction. I remember being introduced to proofs by induction when I was in Junior High. Nowadays, if it is not part of Common Core, it wouldn’t be taught much anymore. Nevertheless, I will use it here to prove that n² = n.
Using a similar proof, we can also prove that n³ = n, n⁴ = n, n⁵ = n, n⁶ = n, and so forth!
Today is the perfect day to review how to use proof by induction so try your hand at proving at least one of those mathematical statements on your own. Use the same steps in my example: prove true for n=1, assume true for n = k, prove true for k + 1, write your conclusion. then have a very Happy April Fools’ Day, Everyone!
Today is also a very good day to review that (x + y)² = x² +2xy + y² and NOT x² + y², a very common error students make. Confession: I remember making that exact error in high school when I definitely should have known better. Using induction to prove something in mathematics is a valid technique, but if you use invalid equations like
(x + y)³ = x³ + y³, you will make invalid conclusions. Thus, today might also be a good day to review the binomial theorem and Pascal’s triangle. (Pascal’s triangle has numbers in its interior, not just 1’s going down the sides, after all.)
My post today was inspired by a post written by Sara Van Der Werf titled Why I’ve Started Teaching the FOIL Method Again. In her post, she not only plays a great April Fools’ joke on her readers, but she explains a tried and true way to multiply binomials and other polynomials.
I read her post exactly one year ago today, and since then, I have been waiting for April Fools’ Day to roll around again so that I could share this post with you. It is my hope that you will enjoy my little prank and learn a little mathematics from it as well.
Now I’ll write a little bit about the number 1371:
1371 is a composite number.
Prime factorization: 1371 = 3 × 457
1371 has no exponents greater than 1 in its prime factorization, so √1371 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 1371 has exactly 4 factors.
The factors of 1371 are outlined with their factor pair partners in the graphic below.
1371 is the hypotenuse of a Pythagorean triple:
504-1275-1371 which is 3 times (168-425-457)
OEIS.org informs us that 1² + 37² + 1² = 1371, and there’s no April Fooling about that!
In elementary school, we learned about improper fractions. Should we call them that? Is it even possible for any kind of number to be IMPROPER? They are simply fractions greater than one. I’ve recently heard the term fraction form used, and ever since I’ve made a point of saying that fractions greater than one are in fraction form.
On Twitter, I’ve found a few people who also don’t like using the word improper to describe any fraction.
This first tweet has a link explaining why it is improper to use the term improper fraction:
Whether it is an improper fraction or mixed number, terminology in maths matters just as much as it does in English, writes Kevin O’Brien https://t.co/B9ZuQThUqf
I always wanted to analyze the "behavior" of any fraction that was called improper! Should this fraction receive some sort of penalty for their deeds? Seriously, knowing their equivalence and when one form may be more appropriate use-wise, is the issue: whether 5/4; 1 1/4 or 1.25
In my 3rd grade class we had a conversation about the term "improper" and how it doesn't fit the fraction. The kids all agreed that fractions can have many different representations and there's nothing "improper" about that. They were super cute.
Does the term ‘improper fraction’ lead to misunderstanding?Does it suggest that a /real/ fraction is less than 1?My goal is to use the term ‘rename’ rather than ‘convert’. We aren’t changing anything but the way it looks. #TVDSBmathpic.twitter.com/jlKKx8uN7l
By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!
If you need some tips on how to get started on this puzzle, check out this video:
Factors of 1350:
1350 is a composite number.
Prime factorization: 1350 = 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
The exponents in the prime factorization are 1, 3 and 2. Adding one to each and multiplying we get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350 has exactly 24 factors.