Vertical Value: Part 1
By Cathy Yenca, posted February 1,
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
Cathy Yenca, firstname.lastname@example.org,
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.