# 9. Four Fours

Highlight: We got student-supplied answers for 1-100.

Lowlight: Those 100 answers did not come from 100 different students. In fact, one student was able to come up with about a third of them himself. We got some but not nearly enough female representation. There was no faculty representation.

Highlight: For the first few days, students were really excited and interested. Their submissions were a bit overwhelming to keep track of.

Lowlight: That engagement level disappeared entirely after a few days. We had 30 or 40 left to figure out, and only 1-2 students were working on it.

Highlight: The leaderboard. Some students really took pride in having their name publicly displayed.

Lowlight: The start of the school year was so crazy that this was the only problem that we had up for the first month.

Highlight: There was a nice unintentional synchronicity between this task and my class. In precalculus, we covered piecewise and step functions, and a couple students noticed that this allowed for crazy new opportunities in this problem. Now they could write 0 or 1 with only one 4 using decimals and the floor/ceiling function.

Lowlight: It was unclear whether or not using e was a violation of the rules. Generally, I like this ambiguity – it allows me to ask the student what they think it fair. Unfortunately, they decided it was fair, and they used it over and over again with the floor function, making things just a bit too easy. In hindsight, I should have pushed him to realize that this shouldn’t count in the hopes he would persist and find a more interesting solution.

The final product (click to enlarge):

What would we change for next time?? Maybe it would be more interesting to focus on a smaller batch of numbers and try to find as many ways to write them? Perhaps that would lead into questions about what makes two equations “different” or not? Maybe a leaderboard with a bigger font so student names are even more public? We definitely need some new ways to publicize our problems and offer different ways for submissions …

# 13. Thirteens

Source: NRICH Maths – Elevenses

Why did we choose this problem?
Good Play With Your Math problems:

• involve some sort of “play” before choosing a specific strategy.
• have a low floor (accessibility and entry point)
• have a high ceiling (need for more complex mathematics)
• have a succinct, accessible, intuitive wording and visualization

This problem meets all four criteria:

• You have to experiment, observe patterns, struggle, and adapt before finding a more elegant strategy. For us, this is what mathematical play looks like.
• This problem is extremely accessible, requiring just integer addition and division in order to test each pair.
• The need to organize the results and systematically test all potential options is no easy task, creating a need for certainty and the desire for a more efficient and elegant strategy. While this problem did not prove all that difficult for me or Ms. Yu, we conjectured that the solution strategy would prove particularly counterintuitive to our students.
• This problem was already succinct and even presented as a poster. We knew it would be possible to present this problem visually, we just needed to decide how much to change.

Modification #1: 13 instead of 11.
We opted for the Thirteens version of the problem for a few reasons: because it was time for Play 13, because we released it on Friday the 13th, and in order to make the problem more challenging. We included the Elevens version on our handout as a scaffold.

Modification #2: Ditch the table.
There is no apparent reason why the numbers should be organized in a square grid. Is this intended to help someone solve the problem? Or might it distract people from the essential features of the question? Instead, we opted to organize the numbers haphazardly.

Early Highlights

• One of my students took one look at the problem and cleverly asked if he could use the 13 at the top. Then we looked to see if it would make a difference.
• I had a Problem Set work day in one of my classes on Friday, and a few students worked on this problem for the entire period.
• Several students kept track of their answers by connecting the circles with a line; however, this became difficult after a while. Perhaps we could have made this easier by arranging the numbers in a circle … or maybe this is the perfect result, creating a need for systematic counting and organized results.
• On a similar note: Ms. Harding solved the problem and organized her results in a much different way than I did.
• One student found all but one of the pairs. When I asked him if he thought he had them all, he said he thought he did, but he wasn’t sure. Exactly!

# 12. Space Race

On Friday, we shared Play 12. Space Race with our students.

Here is a sketch of how this problem developed.

1. Find a fun problem.
We met Ben over the summer at a BU discussion workshop, showed him some of our Play problems, and talked about our approach to the project. Later, Ben sent us a problem that he thought would make a good Play problem. The problem was of his own creation, and had been posted in Math Horizons (April, 2011):

April’s Sandbox problem comes from Benjamin Dickman of Columbia University Teachers College. On the first day of math class, thirty-six desks are arranged in a circle for eighteen boys and eighteen girls. The teacher says that students can take turns choosing their own seats as long as the gender of the students alternates between turns.The last two students to pick will be Amanda and Bill (not necessarily in that order).

Bill has a crush on Amanda and is determined to sit next to her. Amanda, on the other hand, can’t stand Bill and is determined not to sit next to him. All the boys want to help Bill, and all the girls want to help Amanda, which sets up Problem 258, You Can’t Please Everyone: which gender should pick first, and why, if the decision is up to Amanda? If the decision is up to Bill?

Ben added two suggestions: 1) that we could choose a smaller number of desks (e.g. 12), and 2) rephrase the context to get rid of the gender elements.

We played around and solved the problem ourselves, and we agreed with Ben – this was a fun problem that our students would enjoy … if we could make it accessible.

2. Re-write the problem so that it is visual and concise.
This problem was wordy, and for a long time, it sat on the shelf as a candidate that wasn’t yet ready to become a Play problem. I took forever to wrap my head around how to adjust this problem, but one day, inspiration struck: it’s a game. Bill and Amanda both want to “win,” but they have different criteria for success. And with that, Amanda and Bill became Player A and Player B. Desks became squares. Students became X’s.The rectangular shape ensured an even number of “desks.” Stripped of the gossip-filled classroom context, it retained the same mechanics and underlying mathematics.

3. Design a draft
I sketched a draft on paper, CiCi (aka Ms. Yu) masterfully brought it to life in InDesign, and we sat down together over February break to tinker and refine.

4. Debate every word
We tried out several different versions of the wording of the problem. Are the squares “boxes” or “spaces”? What should the title be? Are the last two spaces “adjacent” or does that vocabulary need defining in order for the problem to remain maximally accessible? How do we phrase the prompt at the end to show that this is not simply a game, but a problem that can be solved?

5. Design a handout
When we sent the draft back to Ben, he misinterpreted what the X’s in the board meant. The X’s are intended to show a game currently being played. He thought that they represented the starting board, adding a new constraint to his original problem. We made a handout so students could play with the problem, and made sure that the board was blank, hopefully erasing the potential confusion.

On the back, we initially planned to add a smaller version with only 12 spaces as Ben had originally suggested, but that only takes up so much space. CiCi did some quick copy/pasting/resizing and all of a sudden we had a beautiful variety of playing board options.

6. Print and give students a chance to play
We posted it on Friday and used it in our classes as a Friday Warm Up. It wasn’t easy to make them stop so I could start class. I encourage you to print out the board and play along with us!

# 5. Maximaze

Source: Math Arguments 180 – Day 24: Arithmetic Challenge
We tried this problem a few times and thoroughly enjoyed it. We created our own maze, took the operation-number pairs from the original version (Copyright(c) 2003 Ryosuke Ito), added a few extra pairs to fit our larger maze, randomized the operation-number pairs, and started brainstorming strategies.

Highlight 1: Handouts Everywhere
Without a handout, this problem is completely inaccessible. In fact, anyone reading this post should probably print out the maze so you can try it yourself. Anyway, we made a second version of the maze to fit a half-sheet of paper and printed a bunch of copies.

We printed handouts on both sides of blue paper and put them in a folder in each of our classes. In the hallway, we had the poster, a folder, and a leaderboard that we updated everyday as needed.

I gave them to all my students, and some of them kept going back for more. Eventually, we started seeing them everywhere. I had students working on it during lunch and after school. I saw teachers trying it out. I found copies on the floor, in the trash, and outside on the street. It rivaled Pentagram in popularity and felt even more satisfying.

Again, I find myself wondering: what made this problem a hit, and how can we replicate that engagement? There were two interesting ways in which this problem different from Pentagram:

• While Pentagram had a quick and easy feedback loop, feedback can be slow for Maximaze. Many students struggled to accurately calculate their scores, and in all honesty, we struggled to calculate them too. Ms. Yu had a friend write a program to speed things up. But once they calculated their score, the social aspect of the problem and the public leaderboard helped show students that there was room for improvement.
• While Pentagram plateaued with repetition, Maximaze got better and better – or at least it did for me. When my initial score was beaten, I went back and tried to be systematic and thorough. I made some conjectures, broke the problem up into pieces, ruled out some wacky ideas, and refined my strategy. And this maze is complicated enough that I was able to make a lot of interesting progress without completely solving it.

Highlight 2: Not Being the Answer Key
Because we created the maze ourselves, we had no idea what the maximum score was. As such, we got to play too. We even busted it out at the restaurant during April vacation. Appetizers and math problems – a perfect combo!

Highlight: Highest Score
Lowlight: Second Highest Score
Highlight: Highest Score
Lowlight: Second Highest Score
Lowlight: Third Highest Score
I’m competitive. I got a pretty high score on my first try, only to be beaten by Ms. Yu. I put some real effort into it and beat her score, only to be beaten again. In the end, she scored 3226 and I scored 3190. I lost by 36 points. And then one of my former students scored 3226 too. Let’s not dwell on this. But seriously, it was fun to work on the problem along with the other students and to talk to them about it on equal footing.

Lowlight: How It Works?
After about a month, the leaderboard stabilized, and all ten names were male. There was initially one female student on the leaderboard from my honors precalculus class, but her best score was the one she got during class – she did not get hooked and keep trying.

While math has historically been dominated by men, that should not and need not continue. As a teacher, I have the power to perpetuate inequity, and if I didn’t stop and reflect on these results, I worry that I might be doing just that.

Are male students better at this type of problem than female students? No. But then why did the male students get higher scores? My conjecture: they spent more time on it. And that thought raises new questions: What makes math problems engaging for all students? How do we create a class/school culture in which all students think they should and can be on the leaderboard?

# 2. Pentagram

Highlight 1: 25 Solutions
21 students, 3 math teachers, and 1 assistant principal submitted successful solutions.

Highlight 2: I was one of them.
I first saw this problem in grad school, and after trying it for a while, I got rather frustrated. My classmates were trying it too, and I wasn’t entirely convinced that the problem had a solution. After trying the problem a couple of different times and taking up several sheets of paper, I gave up and forgot the problem. I remembered the problem when we were starting Play With Your Math, and I solved it in about 20 minutes. It was incredibly satisfying to solve it 18 months after failing.

Highlight 3: Amazing Participation
This problem got an even better response than the first one. This problem is magically engaging; all sorts of people are willing to try it and it is great, twisted fun watching people struggle with it. In addition to my students and colleagues, we’ve gotten friends and friends-of-friends to try this problem, and they all get hooked. Some people get hooked and then get angry at us for showing them the problem.

This engagement makes me wonder: Why do so many people like this problem? Can I replicate that engagement elsewhere?

Dan Meyer was asking these questions in April when he brainstormed a list of criteria for Tiny Math Games. His criteria is below in bold, with my reflection unbolded afterward.

• The point of the game should be concise and intuitive. Check.
• They require few materials. Just paper and pencil. Check.
• They’re social, or at least they’re better when people play together. It is not as obvious that this problem would be more fun with other people, but it definitely was. We posted this problem on a Friday, and I had 5-6 students hanging out after-school until around 5:30 PM working on the problem. Two of the students solved it, and their celebration and subsequent trash-talk was priceless.
• They offer quick, useful feedback. Simply count the triangles, but be wary of the dreaded 4-sided triangle.
• They benefit from repetition. Check … to some degree. After a while, it can feel like you’ve tried everything.
• The math should only be incidental to the larger, more fun purpose of the game. meh?

Highlight #4: Persistence
I find this problem to be somewhat paradoxical, in that this problem is:

1. endlessly frustrating
2. totally doable

As a result, people never truly give up. I have had people tell me months later that they are still trying. One of my colleagues is one of them (you can do it Ms. Harding!).

I should mention that the “totally doable” part of the paradox is debatable. I was taken aback when I finally solved this problem because it didn’t feel like I had really made a discovery nor had I shifted my thinking in any meaningful way. But one of my colleagues disagrees; she is adamant that the problem requires thinking “outside the box.” So I’ll finish with a follow-up question for anyone who solved the problem: Do you think the solution requires thinking outside the box? How so?

# 1. Split 25

The problem that started it all! This problem was a huge hit with our students, and we actually got enough submissions to sort through them, determine the winners, and post their solutions for all to marvel at.

Important note: I suggest that you stop here and try the problem if you haven’t already. The solutions below could be considered spoilers.

Highlight #1: Participation
The scope of participation on this problem was truly incredible. I tried it out with all of my classes, most of my students were engaged, and a few were hooked. But that’s not all. Somehow, the problem ended up in an all-staff email and on the school website. We had participation from students who didn’t have any of the teachers involved in making it. A handful of non-math teachers approached me to ask about their progress, plus a couple of administrators. Our physics teacher provided one of our winning solutions. She maintains that she could have done better, but the math teachers spoiled it for her.

Highlight #2: Variety of Solutions
I love how this problem is open for interpretation. What is a piece? Does it have to be a natural number? Can it be negative? Can it be a fraction/decimal? Can it be irrational? Each interpretation has its own best answer, and our students found just about all of them. Our second winner was a group of students who realized you can make the number infinitely big if you allow negative numbers. I was really impressed by their explanation and their attempt to generalize, especially since these students have probably never made this type of analysis before.

Highlight #3: Learning new math.
One of my honors precalculus students – and our final winner – took this problem to its most interesting conclusion: e. The number of steps he took to reach this solution is really impressive: he found the maximum with natural numbers, realized he could use negative numbers, concluded that wasn’t all that interesting, realized he could use rational numbers, created a formula, challenged his formula, drew a graph, analyzed his graph, made a conjecture, improved his formula, and even indicated the domain to which it applied. If I could only get him to put this much effort into his homework …

Highlight #4: Intellectual Vindication
As we were waiting for student solutions, we saw this problem was in the March 2014 edition of The Mathematics Teacher, in “A Rationale for Irrationals: An Unintended Exploration of e.” An interesting title for this problem, considering many students did great mathematical thinking and only one found his way to e, but clearly we were not the only teachers to love this problem.

Room for Improvement: Many students tried the problem for a little while, settled on an answer, checked with someone else, realized they could do better, didn’t immediately see how, and stopped. We have work to do when it comes to mathematical perseverance.

# 0. Origin Story

Step 1: New Year, New Culture
Entering my second year of teaching, I had no idea what our department culture would be like. We lost several seasoned veteran teachers and replaced them with teachers who were somehow even younger than me. I had discovered math-ed blogs over the summer, and throughout the fall, our conversations became more and more influenced by posts of the MTBOS. And then we took things one step further: a few of us started doing math together. We started sharing cool problems with each other, working on them together, and comparing our solutions. The Mathematics Teacher’s calendar problems was our main source of material along with Math Arguments 180. The problem that really stands out in my memory is Breaking Twenty-Five – a problem that traced back to Dan Meyer and Malcolm Swan as well. We had a blast playing around with that problem.

Step 2: A Great Idea
One of my colleagues – Ms. Yu – was inspired, and wanted to share the fun we were having with her students. She made up a poster, and emailed me a copy. The poster had the “Breaking Twenty-Five” problem, some great stuff about how she wanted the “best” answer and not the “right” answer, and logistical stuff about due dates, extra credit, etc. But what struck me was this passage:

“When I was a kid, my mom always told me to stop playing with my food. I didn’t really understand why—I learned so much by playing with my food! For example, I learned that mixing green and red ketchup doesn’t get you purple ketchup. And that cheese doesn’t go well with most Chinese foods except crab rangoon. And aluminum foil in the microwave, well… let’s just say that when you play, you learn.”

(Another great example of intellectual vindication.)

Step 3: Blow It Up
I saw the poster, and I was so impressed that I wanted in. I suggested we make this a department wide project, and we decided to make a new problem every couple of weeks with posters in our classrooms and in the hallway. I imagined our students like little Will Huntings, stopping in the hallway because they saw a cool math problem and unable to walk by without trying it out first.

Step 4: Rollout
We spent most of February break working on this project – finding and trying really awesome problems, discussing the criteria for a good “Play” problem (the entry point is the key), and working on the design. In the process, I learned a ton about accessibility, engagement, collaboration, and how to use InDesign. All of the planning and preparation went better than I could have ever hoped, and after having a few colleagues provide feedback, we launched Play With Your Math at the beginning of March.

Stay tuned to hear about how it went.