20 x 9 Challenge

Today I’m excited to share the 20 x 9 Challenge with you and your students.

Last summer, I picked up a copy of Pierre Berloquin’s 100 Numerical Games at Goodwill for a couple of dollars.

It’s turned out to be a good investment because it has served as inspiration for several fun classroom challenges.

Want to see if the book has anything you could use in your own classroom? If you have an Amazon account, you can use Amazon’s “Look Inside” feature to read some of the puzzles/games in 100 Numerical Games for free.

Just keep clicking “Surprise Me!” to see more puzzles!

I turned several of the puzzles in this book into the Twos to Nines Challenge which I ended up using last year on the first day of school.  So many of you readers have used the challenge this year with your students already, and I have loved seeing all the pictures!

Another puzzle in this book caught my eye recently, and I decided to turn it into a classroom challenge as well.

Here’s the 20 x 9 challenge puzzle (or game as the book calls it) as originally written:

I have dubbed this the 20 x 9 Challenge since it requires you to find 9 different ways to write an expression that equals 20. Sounds easy, right?

The challenge is complicated by the fact that each expression can only use a single digit, and that digit can be used no more than six times.

Here’s the template I ended up creating to give to my students for the 20 x 9 Challenge.

You could print individual copies of the activity for students to write on or you could print just a class set to be used in conjunction with dry erase pockets.

If you don’t have a classroom set of dry erase pockets, you could also use heavy duty sheet protectors. But, I highly recommend investing in a classroom set of the pockets since they are so much more durable.

Yesterday was the first Friday of the month which meant it was Early Release Day. As a result, classes were shortened from the usual 50 minutes to 40 minutes.

My students had a short ACT bellwork quiz to take at the beginning of the class period, so we only had 35ish minutes left afterwards.

This was fine with my Algebra 2 classes, because we were able to use the time to do some dry erase practice with sketching transformations.

My Pre-Calc students, on the other hand, needed to start a new section which required a full review of special right triangles from geometry.

I did not want to tackle this on shortened Friday schedule that also happened to be a Friday where quite a few students were gone and the first football game of the year. And, that’s the story of how my Pre-Calc students ended up being the testers for this activity.

In reality, it’s suitable for students in much younger grades. All students really need to tackle this challenge is some knowledge of the order of operations and some persistence.

They were super engaged by the 20 x 9 Challenge, and I really didn’t even have to do much explaining at all. For the most part, students picked up the challenge, read the instructions, and just dived in.

The main questions I ended up answering were in relation to whether certain things were allowed. The instructions don’t specify what mathematical operators that students are able to use.

Many students wanted to use exponents. I allowed this as long as the exponent was the specified digit AND the exponent counted as one of their digits. For example 3^3 was fine, but that counted as using two threes. I also had students using square root signs and decimal points.

One group even ended up using a vinculum for a repeating decimal which I found to be quite an interesting approach. Of course, one could argue that this meant they were actually using infinite digits…

When I posted the challenge on twitter, many people were using factorials in their solutions. Factorials don’t show up anywhere in the Oklahoma standards, so none of my students ended up taking this approach.

I also allowed concatenation, and I believe that you have to allow this for this puzzle to be possible. For example, two 2s could be put together to form 22.

Really, it’s up to you to make up your own rules for what you want students using and not using in this challenge.

As part of the process in writing this blog post, I took a look at the solutions provided by the book’s author. Each solution is possible WITHOUT any factorials, square roots, exponents, or decimal points. (Okay, the author does post a solution involving an exponent, but most of my students found a simpler solution for that number which did not use an exponent.)

Each solution is possible using only concatenation, addition, multiplication, and division. Oh, and parentheses, of course!

I’m intentionally NOT posting my students’ answers to the 20 x 9 Challenge to this blog post because I already had one group of students go googling for this challenge on their chromebooks in search of the solutions. I know it can’t just be my students that do that…