## Subtraction Towers - Fay With Your Math (#1)

My school has a wonderful math culture. Students are excited for math class and many enjoy solving problems and puzzles.^{[4]}We are not perfect of course, but we have a strong foundation. I always wish that I had more tasks for them to work on.

In the past, my school has had monthly math problems for students to work on independent of their course work, but this practice has fallen by the wayside in recent years. This spring, I discovered Joey Kelly and Xi Yu's Play With Your Math problems and was inspired to make my own version: "Fay With Your Math".^{[5]}I got Joey and Xi's permission to use a similar format and to use my school's name as a pun of their title. My students really liked their problems and will definitely recognize the reference. I set a goal of preparing nine problems this summer. I will share them on the blog as I finish them.

**Mathematics**

I looked for problems that are simple to explain and understand but are more challenging to answer. I am particularly interested in questions where the challenge is in determining when all solutions or possibilities have been found or when the goal is to maximize or minimize. These questions tend to have a wide range of access points (you can find 1 solution) and a high ceiling (you can reason about ALL solutions/conditions).

These questions are designed for students in grades 3 to 6 -- skewed towards 5th and 6th graders. In the past, parents and older students have solved similar challenges as well, and I am hopeful that will continue.

**Motivation**

There is no academic requirement to try these problems and there will not be an academic reward for completing them. Solving math problems is fun on its own! Not all students will attempt these puzzles, but I do not want to set up a relationship where students complete math problems specifically for rewards.

I do, though, want to motivate solvers to provide an explanation for their solutions and the strategies they used to solve each problem. For this, I am going to offer prizes (mostly candy) to the "most-thorough" explanations. I will offer separate prizes for different grade levels and provide guidance on what I am looking for in terms of "most-thorough" explanation.

**Presentation**

I am going to claim a bulletin board for the year and display each question as a poster. I can also use this space to provide details about what makes a thorough explanation and to show off explanations for earlier problems. I will also make a little slot where I can post printable versions of the problems for students to grab.

A long-term goal is to set up a website like playwithyourmath.com, but I'm not going to work on that until I have a bunch of other summer work done. Okay...let's get to the first problem: Subtraction Towers:I chose Subtraction Towers as the first problem because it is quite accessible to younger students. The only mandatory background knowledge is whole-number subtraction. I want to get them hooked early. It seems so easy to find a tower that works, but it can be quite tricky, especially once you try 4 or more rows. I have also used this questions with students in the past and they have enjoyed it very much. So, I wanted to begin with something I feel confident will work.

I also enjoyed revisiting this problem now that I have learned some of the basics of Python. I was able to write a program that can solve this for n-rows.^{[6]}I would be happy to share the program, but I don't want to post it here until I have finished using this question with my students. My students are too good at googling to post a solution on a website linked to my name. Hit me up on twitter (@mathfireworks) if you want to see my solution sooner. Though, the processing time of for solutions with greater than 5 rows quickly jumps into days not hours.

If you want to use this task, here is a PDF with two copies per page. Cut down the middle and have fun!

**Other Uses**

Subtraction Towers is fantastic for getting young students to practice their subtraction facts. While not technically a game and not quite in Dan Meyer's Asprin/Headache categories, it does take a task that is boring (practicing single-digit subtraction) and provide students with motivation for wanting to do it. This might also make a fun first-day task with older students, though I like to have more discussion and group work in my first day.