By Cathy Yenca, posted February 1, 2016 – 

When I first started teaching, I remember planning and designing lessons for topics I had never taught in my life. I knew early on that my heart belonged to middle schoolers. After taking college courses like Differential Equations and Discrete Structures, I faced new and scarier challenges, like how do I teach middle schoolers to understand ratios? Reading Mathematics Teaching in the Middle School was a lifeline for me when the only teaching experiences I could draw on were other professionals’ articles and lesson ideas.

Over time, I found tasks that have what I call “vertical value”: In broad brush strokes, these tasks can span multiple grade levels and can be used as tools to elicit student misconceptions.

I found one such task that helped students understand concepts of perimeter and area and how they can operate independently. Giving the task to seventh graders, I thought it would be a quick “warm-up.” Little did I know that this simple group of rectangles would bring forth misconceptions and facilitate lots of good discussion—hallmarks of tasks with vertical value.

A surprising number of students miscounted the perimeter lengths, finding 16 centimeters instead of 20 centimeters when they turned the corners. Many students erased and changed their perimeter answers, thinking that they were wrong when they got the same perimeter repeatedly. Students’ eyebrows illustrated consternation as students found that perimeters remained constant but areas did not. How could this be? Follow-up discussion questions clearly got students’ wheels turning, convincing skeptics that the perimeter does not change with the area after all. After such great conversation, I was proud that I had taken the time to cut out and tape rectangular grids to that activity sheet. This task has remained a mini-lesson in my classroom throughout the years.

Stay tuned for part 2 of this post where I will share how transforming a simple concept from an activity sheet to a Desmos activity can further increase its vertical value.

Cathy Yenca, cathy@mathycathy.com, is a math teacher in Texas. She tweets at @mathycathy and blogs about teaching mathematics in a one-to-one iPad classroom at mathycathy.com/blog.