I was working with a 7th grade class on the Triangle Inequality Theorem.  Groups were given 8 pencils from 1 in length to 8 in and were asked to create triangles given various combinations of pencil lengths.  Some combinations worked whereas others didn’t.  The exploration led the students to The Triangle Inequality Theorem.

The activity used, Triangle Inequality Theorem – Investigation, Guided Notes, and Assignment , was downloaded from Teachers Pay Teachers.

Throughout the day a few conversations and observations stood out.

Example 1:  Playtime

As I approached one group, I hear lots of discussion, debating really.  In an attempt to settle the debate, Xavier grabs the pencils and said, “Let’s play with the numbers!” He wanted to show his group how to experiment with the pencils (the given manipulative)

Xavier’s comment was music to my ears! Manipulatives give students a chance to play with the math.  For many students, the tangible object clarifies the concept therefore building a deeper understanding of the math.

Example 2:  Spacial Visualization Confusion

Students require opportunities to strengthen their spacial visualization skills.  One group was absolutely convinced that the 3 pencils (6, 7 & 8in length) would not form a triangle.  I recreated their first attempt (bottom left) for I didn’t take a picture at the time. I suggested to move the pencils. 3 students removed the pencils and then slid them back into the same spot:).

Surprisingly, this scenario happened multiple times.  I responded in a few ways:

• Have you tried changing the angles? (When that didn’t work)
• That combination does create a triangle.  Now prove it. (When that didn’t work)
• Without giving them the answer, I showed them what I meant by changing the angles.

Once comfortable manipulating the pencils, student effortlessly worked through the task.

Example 4:  When the rule gets in the way

• Me:  Does this combination create a triangle.
• Group:  No
• Me:  Explain why
• Austin:  The sum of the 2 smaller sides are less than the third.
• Me:  (Knowing that this students  attends a math program outside of school)  True.  How is that rule displayed in the picture.
• Austin:  The sum of the 2 smaller sides are less than the third.
• Me:  I’d like you to apply that rule to this picture.
• Austin:  Stares at me
• Me:  Yes, the sum of the 2 smaller sides is less than the largest side.  What happens in the picture when that rule is present.
• Austin:  There’s a gap.
• Me:  Yes!  And why is there a gap?
• Austin:  The sum of the 2 smaller sides is less than the largest side.

In this situation, the teacher had been out for several days due to a medical need. Because of the rotating substitute teachers, assessing student understanding was necessary before moving on.  I knew Barry’s background therefore not surprised that he knew the rule. I wanted him to apply that rule to the given picture.

By moving fluidly between the abstract (rule) and the concrete (visual), I strive to strengthen flexible thinking within my students.  Here are  2 conversations designed to illustrate my point.

Example (not an actual conversation)

• Me:  What does your picture tell you?
• Student: It’s not a triangle?
• Me: Why?
• Student:  There’s a gap or There’s extra
• Me:  The gap is describing a rule.  What is it?
• Student:  The sum of 2 sides is less than the 3rd, therefore the sides don’t form a triangle.

In reverse:

• Me:  What does your picture tell you?
• Student: It’s not a triangle?
• Me: Why?
• Student:  The sum of 2 sides is less than the 3rd
• Me:  Looking at the picture, where is that rule displayed?
• Student:  The gap or  In the extra

Closing  Thoughts 

Throughout the exploration, which lasted roughly 12 minutes,  I found myself observing groups who were completely engaged.  Although the activity was structured in a cooperative group format, where all students had an integral role within the task, there was at least one student per group who relied on moving the pencils to further their understanding.  For others, their eyes were glued to the picture.  Seeing is believing and the examples of triangles and non-triangles formed a lasting reference.

I’ve always been a tactile/kinesthetic learner.  Still am. I learn best when I’m actively involved in the process.  I would have been the student who relied on manipulating the pencils to see the answer.  When students are playing with manipulatives, they are learning.  In many instances, students are learning more than expected and these unintended yet beneficial connections develop a student’s capacity to understand mathematics.