Rational Expressions – Count On It


Global Math Week!! 10/10/2017

Today I had the pleasure of serving in my official role as an ambassador of Global Math Week. More than 910,000 students participated from around the globe, and the lesson was one that had applications for all ages.

The lesson was the idea of Exploding Dots. Rather than explaining it, I’ll share this link with you, since the creator, Dr. James Tanton, explains it much better: https://vimeo.com/204368634

I worked with our district’s Math Director, Mary Kemper, to visit as many campuses as we could to see how they were teaching and applying Exploding Dots in their classrooms.

Our first stop was Lee Elementary, where educator May Voltz had clearly been teaching this for a while. Students began with a review: Several machines of different bases were drawn on the whiteboard with dots already filled in. The students were required to figure out the value in each machine. Students shared their answers with the class by writing on the whiteboard.

Students were then given four new questions, again with different machines, and dots filled in. Each question was outlined in a different colored square. The question was, which of these values was different from the others, and why? Students made short work of calculating the values. But then they had to compare and contrast with the other answers to see which was different. Once the choice was made, a student could stand under a colored ball hanging from the ceiling. The color of the ball of their choice matched the color of the machine value that was different from the rest.

As you can imagine, there was reasoning why any of the four answers were different, and none of those reasonings were incorrect. Students were successful by simply having a justified opinion.


photos by Ian VanderSchee

I then ventured to Middle School East, where several teachers were participating in this event. A presentation had been made with animations and sound, and shared with all teachers.

Mary and I were was able to visit the classrooms of Amanda Cooper, Brooke Apple, Leia Poskey, Shawn Reck, Donna Arnold, and Kindal Renaud, all of whom shared their enthusiasm for Exploding Dots with their classes. It was quite an inspiring sight!

All of the teachers were starting at the beginning – explaining the “2 to 1” machine. Since these were middle schoolers, they learned how to apply it to long division. Students learned how to group dots together according to what the divisor was, and they discussed what to do if there were leftover dots that couldn’t be grouped. It was a problem solving activity, but most if not all students agreed that this method of dividing numbers is much easier than the old way.


photos by Mary Kemper

What a rewarding day!! I can’t wait to get back to my classroom and work on Exploding Dots in my own classes. Now that I’ve seen the excitement it created, I look forward to seeing that same excitement from my own students!!

Rational Expressions – One-Liner

Rational Expressions – Blaise of Glory

Rational Expressions – The Path Not Taken

Rational Expressions – Elementary

Rational Expressions – Exponential Decay

Rational Expressions – That About Summers it Up

I guess this only works if you measure temperature in Fahrenheit and angles in degrees. Anyway, …

Rational Expressions – Branching Out

Flipped Classroom – First Five Days: Grads Talking to Seniors

Last year, during the first five days of school, I invited my former students to visit my classroom, either in person or through Facetime, to talk to my current seniors and give them encouragement and advice. The response was overwhelming! The graduates were eager to share their experiences with the new seniors. Even though the graduates had schedules of their own, I was able to organize a schedule where each class of seniors was able to hear from one or two students each class, for the entire week. I was also able to make sure that the seniors were able to hear from students from two distinct groups: (1) recent graduates and (2) graduates from over a year ago.

Recent graduates proved to be very helpful with the college application process – which schools to apply to, financial aid, scholarships – as well as the stress of finishing all of the work required to earn their International Baccalaureate (IB) diplomas – multiple internal assessments, papers, and an extended essay.

Graduates from over a year ago were extremely helpful with advice about the college experience after admission – attending classes, getting involved in campus life, etc.

The event was so successful, and the feedback from the seniors was so positive, that I decided to do it again this year. Once again, the response was incredible! Personally, I smile because I’m able to see my former students again. For the students, the advice they received was so valuable, even though it was scary at times. There was a common thread to all conversations, with small differences in the details.




In general, the seniors were given the following advice:

  • Submit everything before the deadline. Any college-related application that was submitted on the deadline would greatly reduce your chances of getting accepted. Any IB work should also be done before the official deadlines so that it’s not hanging over your head.
  • Apply to a variety of colleges – “reach” schools, state schools, and “safety” schools, knowing that even if you don’t get into your dream college, you will still have a successful and meaningful college experience.
  • Enjoy your senior year – this is the last time you will be around your closest friends every day, so enjoy that time and make memories.
  • Actively research scholarships – there are many scholarships out there that people don’t know about. Do your research and apply to the reputable ones. And you can apply for scholarships each year you attend college, not just your freshman year.
  • Recognize that IB is preparing you for college, moreso than other advanced classes – Most of your time in IB is spent doing independent research and writing original reports and essays, so being able to manage your time, balance your activities, and take ownership of your learning in college is easier if you’ve been in IB.

Each graduate then spoke about his or her unique choices, experiences, problems, stresses, solutions, and strategies. They discussed how they narrowed down their college choices, how and when they applied for scholarships, how they strategized to get the most financial aid, what they do with their spare time, when they study, how they deal with dorm life, and so on. The answers varied greatly at this point and the seniors recognized that there are many ways for each of them to be successful at college.

I’m hoping with so many different perspectives, the seniors realize that there is no such thing as the one right path for everyone. I also hope they realize that if they plan and work hard, things will work out. Maybe not the way they dreamed, but they will certainly work out for their success and happiness.

♥ Thank you to Edward, Lea, Sophia, Adithya, Jess, Erin, Bobby, Angie, Jia, Veronica, Maanas, and Pam, for fielding random challenging questions. Video calls, talking to groups, answering questions without the opportunity for preparation – each of these things is difficult, but you handled all of it like professionals. I can’t tell you how proud and impressed I am with all of you. ♥

Flipped Learning Global Initiative

Supporting Flipped Learning Worldwide

Continuous Everywhere but Differentiable Nowhere

I have no idea why I picked this blog name, but there's no turning back now

Learning in Liz's Life

Experiencing, reflecting, growing, and learning

Mathematics for Students

Learning Mathematics - Resources for Students


Ramblings of: a HS math teacher; lover of technology; addicted to learning; passionate about making learning feel like play.

janet's life

growing up and journeying forward



Flipping with Kirch