Math Hombre

Interesting time to be hosting this carnival! I feel like there's a small resurgence with blogging, and I want to be part of it. I've really missed writing informally professionally. I've been a part-time host since Math Teachers at Play 22, 14 years ago!, and a big part of the original purpose of the blog was to collect, curate and share things that delighted and supported me. If you're interested in hosting, contact Denise Gaskins, the founder of this here carnival. The January carnival will be at her blog, but I think February is open!

177 is semiprime, for which two primes? 

177 is the ninth Leyland number, of the form x^y+y^x. What are x & y for 177? They're both prime, which should be a special kind of Leyland I think.

177 is the first "non-trivial" 60-gonal number. (1 and 60 are too easy.) What is the next 60-gonal number? (Pictured) What does the sequence of the first non-trivial n-gonal numbers look like? (6, 9, ...)

177 is a Leonardo number, so named by Edsgard Dijkstra for their relation to the Fibonacci numbers. The first five are 1, 1, 3, 5, 9... can you determine the pattern?

But the coolest thing to me is that it's the magic constant of the smallest magic square of distinct primes! I'll get you started...

 All this month I'll be posting games from the Fall 2023 GAMES seminar at GVSU. This senior capstone was begun by Char Beckmann. See many of the games from her seminar in this YouTube playlist. Many of the games completed in my seminar are in this playlist. In the seminar, we play lots of games and math games, the future teachers make a first video to promote a class math game that already exists, we develop a group game (a monster-themed middle school Desmos escape room and Math Heads, a number mystery game this year), and they develop a game of their own.

Keri Herman chose Tens Go Fish, a classic addition fluency game. As an extra feature, she demonstrates the game with Tiny Polka Dot cards. (Find them here at Math for Love.)

Keri's original game was a new idea for the seminar: she was interested in making a puzzle. I recently saw a description of a puzzle as a game for one person.  That certainly fits here. The puzzles are available on a Google doc here. What follows is Keri's story of making the game, and her ideas on why we should play games in math class.

Story of my Game

I knew when I got the opportunity to create my own game, I wanted to develop something that was related to quick multiplication facts. The reason being that my memory of learning my multiplication tables was always timed and quite stressful for me as a young student. I wanted to create a game where students could get great practice of their multiplication facts, and build in aspects of a good game; strategy, any player can win, etc. 

My first idea was a game board, moving the amount of spaces of the product. However, I was then drawn to the idea of more of a maze. I started with a small grid and filled in very small multiplication facts, students would have to find their way to the end. This turned into the development of three mazes, 5 x 6, 7 x 8, and 9 x 10, all with their own unique solution. To figure out how to design these mazes took a lot of different approaches, starting from scratch, and overall just thinking about how to make them work. I believe that the final product of these mazes will provide students with a very fun way to practice their multiplication, while being able to try to solve the maze. 

The goal was to have a large mathematical objective for the game. Students will be focused on trying to find the solution, even if they are going the wrong way, or have to start over, they are still constantly doing the math and getting practice of their multiplication facts. I think this game would be something that teachers should play with their learners because you can never have enough practice with multiplication. Especially in the 9 x 10 maze, all multiplication facts are used from 1 through 9 (not including zero). These mazes will also help students recognize patterns between multiples, factors, and products. 

These games could be used within a lesson, if students finish early, or simply just given as an opportunity for more practice, without time constraints. I also share within my video the development process of these mazes. With students who have learned multiplication facts,  I think it would be a great idea to turn this into a project or performance assessment. Students can work to develop their own maze. Not only does it take strategy, but at the same time students are able to continue working with the facts themselves and continue to recognize patterns. Overall, I am very proud of the way these mazes have turned out. I want to continue to show these to math educators and I hope that students will enjoy solving them as much as I had hoped. 

Why Play Games in the Math Classroom?

As a future math educator, incorporating games into the classroom is something that I want to use and will continue to encourage others to consider as well. It is often looked over to play games in the classroom, but the reasons as to why they are beneficial to student education should be considered. There are few specific reasons that are important to point out, including; building mathematical knowledge and skills, collaboration with peers, student engagement, critical thinking skills, and more. Each one of these reasons in its own makes games in a math classroom worthwhile. 

Building Mathematical Knowledge

Math games all are built upon their own goals and mathematical objectives. Teachers have the option to choose a game that targets the content that is being focused on. To find a game that can build mathematical knowledge, choosing a game that is relevant to your current learning goals within a classroom can help students extend their skills. There are so many aspects built within games that students can pick up on mathematically, without noticing. This can be beneficial to students because they are still learning, but without the title of class, homework, or assessments. 

Collaboration with Peers

It is important for a classroom to have communication among students that can lead to quality discussions. Discussions can uncover so many helpful aspects to student learning. In a game setting, a lot of times students will play with each other in teams, or against each other. In both cases, students are able to communicate and learn from each other. Students are able to pick up on each other’s strategies and build off of them. When playing with each other, this can help build a more positive classroom environment. This is because this type of communication is not usually seen in a regular lecture or discussion. 

Student Engagement

Oftentimes we hear negative assumptions about math and negative attitudes are common when stepping into a classroom for some students. It is important as teachers that we are able to increase student interest by engagement and participation. Incorporating math games into the classroom is a great way to develop student engagement. A lot of times, the mathematical objective of games are mixed in with aspects of interaction, surprises, and fun. A game can also change the view of many students. All students can participate and it is important to use games where any student can win. In math class, students can often point out the “smartest” students and become discouraged. When using games that are designed that anyone can win, not just based on skill, this can build a lot of confidence in students. 

Critical Thinking Skills

In many situations, students become disengaged after they reach the level of knowledge and understanding. However, it is things like analysis, critical thinking, and application that get students to really push past that level of reasoning for the content that they are learning. Math games provide a different way to push students to build upon their critical thinking skills. Having to figure out a strategy to finish or win the game is a very important tool when it comes to building these skills. With that being said, games that are chosen to play in a class should have aspects that involve strategy. 

Overall, math games have so many advantages when it comes to incorporating them into the classroom. Being able to play different types of games this semester has taught me so much about what a good math game should look like. Being able to develop and create our own group game, and my own game has changed my perspective on math games. Math games can help students learn in a unique, fun, and interactive way. 

 Do you want to host the 13^2 Playful Math carnival in October? A month that had a Friday the 13th? 

Yes, please. Should have been on 10/13 instead of 10/31 but... apologies.

169 is a palindrome in two number bases less than 16. Which do you suppose?

All odd squares are centered octagonal numbers, but 169 is also a centered hexagonal. (Visualize more with Alex CHIK's GeoGebra.)

 I'm teaching a quick 6 week Intermediate Algebra (linear/quadratic/exponential) for incoming freshman this summer. Part of my goal is to convince them that math is different than how they might have been exposed to it. On day 1, we started with Wordle. A few learners had played it before, but quickly the whole class picked up the idea, and there were several good deductions about which letters could go where. The rest of the week, we played the daily Wordle before class the rest of the week.

This week, we started with SET. A little harder to understand, but there's so much logic. The daily puzzle has up to six solutions, which seems to allow for more participation. (Kelly Spoon noted Set with Friends for online actual game play, plus variants.)

I had ideas about what I wanted to do in subsequent weeks, but I was curious what others think and asked on Twitter. BOOM, people exploded with a bevy of resources. I used to have a blog where I shared resources, where did I put that...? After Sam and Julie posted about moving to Mastodon (because of Twitter's Troubles), I tried posting there, too.

Math Online Games & Apps

• FiddleBrix suggested by Benjamin Dickman. He suggested downloading the app, then handwrite a previous puzzle. This is a super challenging puzzle, to me, but Benjamin's suggestions are gold.
• SumIt puzzle suggested by Kelly Spoon. Lots of stuff there.
• Beast Academy All Ten also via Kelly. Really great arithmetic challenge.
• Draggin Math pay app, 
• Shirley McDonald suggested a lot of great stuff: All Ten by Beast Academy (always an open tab in my browser), Number Hive (like the Product Game on a hexagon board), Skyscrapers (Latin square with clues, from a site with lots of puzzles) and Digit Party (implementation of a Ben Orlin game; also an open tab, I may have a tab problem).
• Shirley also recommended Mathigon's Puzzle of the Day. I've been playing that in an app more days than not. (I think I'm getting better?)
• Kathy Henderson suggested the NYT Connections game, which I hadn't seen yet. That is very much in the spirit of what I'm looking for!

IRL Math Games (Free and Commercial)

• David Butler has a great collection of activities, his 100 Factorial. He singled out Digit Disguises and Which Number Where
• Neal W recommended: Quixx is a great dice game and very easy to learn. My students love 20  Express. There are rules and scoresheets online.
• Tom Cutrofello suggested the excellent Turnstyle puzzle he designed for Brainwright!
• Prime Climb by Dan Finkel, suggested by Amie Albrecht. She notes, especially David Butler's human scale Prime Climb. (Which I have played and love.
• Anna Blinstein suggested Anna Weltman's Snugglenumbers, which is a great variation on a target number game.
• Pat Bellew said remember the original: Mastermind. Erick Lee has a Desmos activity implementarion of the math version, Pico, Fermi, Bagel.
• Sian Zelbo claims Jotto is better than the either Wordle or Mastermind. (Online version.)
• Becky Steele cited David Coffey for Taco Cat Goat Cheese Pizza as well as Farkle.
• Chris Conrad recommended Quarto, amazing strategy game. Amie mentioned you can play with SET cards - how amazing is that idea. Karen Campe remembered this great Aperiodical article about the game.
• Mardi Nott, Bradford Dykes and Jenna Laib vouch for Charty Party - that's a strong recommendation. Bradford also brought up this stats version of Spot It, the Graphic Continuum Match It Game.
• Ms. Morris suggested Nine Men's Morris. Interesting game idea.

Puzzles

• Kim McIntyre suggested Sarah Carter's big collection of classroom puzzles. I have learned so many puzzles from her over the years, but especially the Naoki Inaba puzzles.
• Speaking of Japanese puzzles, Gregory White suggests Shikaku.
• Benjamin Dickman and Shirley and Gayle Herrington suggested KenKen. I've used those with younger learners and college students.
• Karen Campe had several suggestions, some in this blogpost. Times UK puzzle page, StarBattle, Suko, 
• suggested Mobiles. Love those, and we do lessons based on them. Here's a challenge problem I asked them!
• Druin suggested the Puzzle Library, which I can't access for some reason. Looks like they're intentionally made for schools.
• Susan Russo linked Cryptograms, which are some cool cruptographic puzzles. I haven't tried anything like this and am curious.
• Sarcasymptote brought up Sideways Arithmetic from Wayside School, which is what I was expecting from Cryptograms, thinking it was cryptarithms. But somehow have never seen that book despite loving Wayside.
• Ms. Morris linked a Magic Square app.

Welcome one and all! Come on in and have a ball. 159 is semiprime and that's just fine.

Lucky that I'm hosting this, or is it just that 159 is lucky. How do you get lucky? Start with the counting numbers. Delete every 2nd number, leaving 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45... odd.  The 2nd number remaining is 3, so delete every 3rd number, leaving 1 3 7 9 13 15 19 21 25 27 31 33 37 39 43 45... now that's interesting in and of itself. Next delete every 7th number, leaving 1 3 7 9 13 15 21 25 27 31 33 37 43 45 ...; now delete every 9th number; etc.  How far do we have to go before we know 159 is lucky? Does knowing 151 is the previous lucky number help? Interesting to look at the gaps in each step, and the cutlist for each step.

Is it rarer to be a semiprime or a number with only odd digits? Odd increasing digits? Linear pattern in its digits?  Alyssa would like it as is.

Pat Bellew's 159 facts are that 159 is the sum of 3 consecutive prime numbers (which?) and can be written as the difference of two squares in two different ways (don't you want to find them?).

He also has that __ __ •159 = __ __ __ __ using all 9 nonzero digits. Of course, you can brute force it, but can you deduce this digitally complete product?

What #playfulmath have you seen this month? Here's some of what I have noticed.

September started with Math on a Stick in full swing. Doesn't get more playful than that!

Katie Steckles and Jimi went over the math in the Spider-Man No Way Home end credits. I lost my mind when watching it in the theatre, and am so glad someone's sharing it. SO MANY MC Escher references.

And as if the visuals weren't sufficient, the song is De La Soul's great cover of the Schoolhouse Rock classic, Three is a Magic Number.

 Welcome to the Playful Math Carnival, 155th edition!

155, tell us your secrets.

Via Pat Bellew, 155 is the sum of the prime numbers between its smallest and largest prime factors, 5 and 31. 5+7+11+13+17+19+23+29+31=155. How would you go about finding more of these? What would you call them? Pat also notes that 155 is the number of primitive permutation groups of order 81. Which is odd, because it is more than double the number of groups for any order less than 81. And there's not another larger (than 75 even!) until you get to order 256 (which has 244). Do 81 and 256 have anything in common?

Wait, 5 and 31? That means 155 is semiprime. What is the previous and what is the next semiprime? (They're both even...) Are there more primes or semiprimes smaller than 100?

The coolest thing I found is that 155 is a toothpick number. You start with a toothpick, then add a perpendicular toothpick anywhere there is an exposed endpoint. Here is 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67. How many more steps to 155? Is it a fractal? Is it a cellular automaton? Mathematicians have also studied T(n)/n^2. Does it have a limit? Does it have an extremum? Here's some GeoGebra to make your own.

155 is also a generalized pentagonal number. The pentagonal numbers have a rule n(3n-1)/2, usually for n =1, 2, 3... , giving 1, 5, 12, 22, 35, ... But there are also positive outputs for negative integers, 2, 7, 15, 26, 40 ... which pleasantly fit between the usual pentagonal numbers. What patterns do you notice? Which negative number gives 155? I've been trying to think about how to visualize these negative pentagonals, to no avail so far. Have you got any ideas?

Maybe the toothpick was a little too crazy of a visual patten? Here's one I was trying to make to have 155. Did it work? If so, which step? Fawn always asks for the 43rd step... what's that? Is there a rule? What if step 1 had -1 square, what would the rule be?

Jennifer Silverman made this cool motions maze the other day. (More here.) We collaborated on it a little bit, after she did all the heavy lifting. I added buttons. (I may have a button problem.) It put me in mind of these motion puzzles I used to make in Geometer's Sketchpad, and I got to thinking how much better I could make them now. So I started, with the main new feature I wanted being the ability to generate new puzzles instead of being one static dynamic puzzle.

xmin=floor(Min[{0, x(F1'), x(F2'), x(F3'), x(F4')}]) - 1
But there's a problem then - the graphics won't be in 1:1 scale, which is always nice, but especially important for motions where the two objects should look congruent!

• xm = If[(ymax - ymin) / (xmax - xmin) < r, xmax, xmin + (ymax - ymin) / r]
• ym = If[(ymax - ymin) / (xmax - xmin) < r, ymin + r (xmax - xmin), ymax]

If the sandals give a screen that's not wide enough, it uses the aspect ratio to find a suitable width. If the sandals give a screen that's not tall enough, it uses the aspect ratio to find a good height. (The If[ ] command works like If[condition, then, else], where the else is optional.

The puzzle, as it turns out, is pretty challenging. Give it a try, and let me know what you think. Or let me know an easier better way to do my GeoGebra graphics hacking.

Here's the teacher page for download or the mobile page.

EDIT: Bonus! Jennifer created an assignment to give it more structure as a lesson. (PDF in dropbox.)