Washi Tape Math

Last week I had a positive experience with manipulatives in both my Algebra 1 and Honors Algebra 2 classes.  This is a shift from previous experiences with manipulatives that have left both me and my students frustrated. I believe the positive experience came from a change in mindset on my part.  Previously, I would have a target that I would want students to know by the end of the 50 min. period.  Within this period, students would work through the activity, but inevitably never at the pace that I had planned.  So, with a few minutes left of class, I'd summarize what they were supposed to learn.  Students would get frustrated because they didn't understand what I was rushing to tell them and I would be frustrated because it felt like 50 wasted minutes.  Because of these experiences, I've mostly done discovery lessons with technology instead of manipulatives.

Dave Sladkey and I worked together to plan a lesson for the target:  Write and apply a quadratic equation from context.  I was struggling with the following practice problem, because I didn't think students would really understand what the expression (3x - 7)m meant.  Dave has been doing work in his classes with blue painters tape and suggested we use tape to construct different sized rectangles on the walls.
We ended up creating this activity where students created a progression of rectangles with side lengths as expressions.  Scott Miller suggested washi tape instead of painters tape because it is thinner and students might struggle with deciding whether to measure from the inside or outside of the tape.  Plus the washi tape made the classroom more colorful!

I invited a few teachers in to observe and give me feedback.  Specifically, I wanted to know if spending the time creating these rectangles was useful and if the questions I asked along the way were effective in guiding and deepening student understanding.  During our post lesson discussion we talked about the slow pace of constructing.  Thankfully Dave and I had planned this to be a two day lesson so that we would not feel rushed to construct, discover and understand all in one day.  Besides pace, the major discussion we had was about the students' rectangles.  They were all different.  We talked about whether I should or should not lead students to make their rectangles in a particular way. In this picture, students had created a 12 in. by 12 in. rectangle followed by a 12 in. by (2*12)in. rectangle.  They were working on the progression to now make a 12 in. by (12 + 3) in. rectangle.  You can see students on the left making a rectangle with a partition inside, leading to the understanding of what a rectangle that is x by (x+3) would look like.  Whereas the students on the right had an rectangle open.  Should I have led this group to represent 12 by (12 + 3) with two rectangles instead of one?  I don't think so, I think that their thinking was the purpose of this activity.  If I had wanted students to think my way, I would have taught from the front of the room.  And this, was my aha moment about manipulatives!  The purpose of doing a manipulative lesson isn't to get students to know something, it is for me to uncover their thinking about a particular topic.

This lesson was slow in a good way.  Students were not writing and solving quadratic equations from a context by the end of the period.  But they were on the second day and we had our work on the wall to reference back to.  In fact, when students walked into the classroom on day 2, they asked to use the washi tape again.  So we did!  This time, students used the tape to create a picture to model this context:  A pool with a uniform walk way around it has dimensions of 12 ft by 20 ft.

This positive experience led me to use washi tape in Honors Algebra 2 to create normal distributions.  My purpose in that class was a little bit different than in Algebra 1.  I had seen Jo Boaler talk at NCSM and was reminded of the interviews she's done with her Stanford students who have experienced math anxiety for many years.  As much as I teach growth mindset, I know there are math anxious students hiding in my honors class.  By constructing with washi tape, the whole class slowed down.  The pressure to finish first, fast or just to keep up was taken away because it naturally takes more time to put washi tape on the wall than to write something down.

A good manipulative lesson has many starts and stops.  Students are sharing their thinking with their groups and with the class.  Different perspectives and representations are shared for the benefit of the whole class.  In the end students were able to see why we model area with quadratics.  That made the whole experience worth it.