Kinesthetic Algebra – Making Abstract Variables Concrete

If you haven’t signed up for Explore the MathTwitterBlogosphere initiative, YOU SHOULD.  Because, as Tina said best,

No matter how much time I spend online, I’m always happening upon new parts of our awesome math teacher community. Even yesterday, when we were planning the first few missions for this oh so exciting event, I learned about some sites I didn’t know existed. This is exactly why I am inviting you to join me on an exploration of the best parts of the internet – the Math Twitter Blogosphere.

Tina Cardone

Yes, this exact same thing happened to me on Sunday.  I got this idea from Emma McCrea.   I tweeted her immediately to tell her how much I enjoyed her lesson, and she tweeted back that she has signed up for Explore MTBoS!  Win-Win!

I gathered spaghetti, yarn, and dental floss cut into different lengths, colored index cards cut into squares and rectangles, foam cubes, and then more of these pieces taped to other pieces.  I dumped them into five bins.  This was actually pretty fun for me.

Notice and Wonder:

When the students came in, I gave each table group a Notice and Wonder form and a bin of stuff.  I told them nothing except, “Fill out the Notice and Wonder form with your group.” We have done Notice and Wonder many times before, so they got right to it.


They were FABULOUS.  Groups had great questions for me such as, “Is this 2D or 3D?”.  They noticed the different size shapes, lengths, and combinations.  They wondered if they were 1D, 2D, 3D or all 3D?  They wondered if they combined shapes were 4D.  Many groups even started sorting them.  Great idea.


I then had the groups sort their objects however they wanted.  I did get a couple of interesting sorts, but almost almost all groups sorted the same way.  So they could see what I was seeing, I had all of the groups walk around and look at all of the sorts.  My crazy sorters were able to see how most of the groups had sorted.



After sorting, they shared their notice and wonder.  I explained to them that ALL of the objects were actually 3D, but we were going to look at them as 1D (length), 2D (flat paper), and 3D (3D objects).  At this point, the kids all said, “Like Flatland!” and a connection was born.

After that I drew pictures of the objects on the board, starting with 1D and moving up to 3D.  We talked about how the lengths of all of our green string were close, but different sizes, and how we can use a variable to name an unknown length.  To make this more interesting, I let them name the unknown lengths.  This was a HUGE hit, and they picked funny names like Wormy (w) and Doofy (d).  As we picked more lengths, it was a great chat when they wanted to name a new length Wanda (w), but could NOT because we already had a w and it was NOT the same length.  It was a blast!

When we had progressed all of the way to the cubes, the kids noticed a pattern.  They noticed that the variables for the 1D objects had 1 as an exponent, some 2D objects had an exponent of 2, and some variable of the 3D objects had an exponent of 3.  I love it when magic happens.

To end up the first day, we discussed the difference between wormy + wormy and wormy times wormy.  I held two wormy strings together to illustrate addition of two lengths.  I also drew a 2×3 multiplication array to illustrate multiplication and make sure they were getting the difference.

They seemed to get it (and seemed to enjoy it).  Tomorrow, I am going to let them cut their own lengths from spaghetti and draw their own 1D, 2D, and 3D shapes.

The next day, I will do Emma’s great sorting activity!  It is at the bottom of the page.  I will let you all know how it goes!

7 thoughts on “Kinesthetic Algebra – Making Abstract Variables Concrete

  1. I love the way you’ve taken the idea and developed into something bigger and better. I’m looking forward to experimenting with ‘notice and wonder’, what a fabulous way to shift learners focus away from just finding the answer and push them towards generating their own questions. Brilliant to read your experience – thanks for blogging 🙂

  2. Pingback: My All Time Favorite Middle School Math Lessons | I Speak Math

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