Solve Me Mobile Equations

 I love using Solve Me Mobiles with my students especially as we work on solving equations.  This year I decided to dig a little deeper into the mobiles to see how I could facilitate more discussion around the thinking used to solve mobiles (which students always engage in) and connecting it to the algebraic work required in class (which some students don't understand or engage in).  

The Open Up Resources curriculum has some great lessons that use the hanger model. If you haven't already you should check them out. Along with that I found the book Making Sense of Algebra on my shelf.  Probably something I bought pre-Covid thinking I would have time.  It is a great read.  It focuses on Habits of Mind, much like math practices, that should be developed so students are successful in Algebra.  One of the chapters, Solving and Building Puzzles, discusses the Solve Me Mobiles.  The book explains that puzzles help build stamina because of the little wins that students get along the way.  That explains why students are more likely to engage in the Mobiles than in an algebra problem. The mobile is easily accessible to all students and those with low math confidence don't look at it and immediately shut down like they might be inclined to do with an algebraic equation.  The mobile allows us to transition the students to the algebraic notation that is presented in the mobile.  What really resonated with me when reading was the statement that we need to make the logic explicit.  Students are doing the thinking, but how can we take a step back and really help them think about what they are doing and how that thinking is algebraic in nature.  

So these were my 2 goals:

1. transition to algebraic notation

2. draw attention to the logic and reasoning being used

After using the 6th grade Open Up Curriculum with hanger models, I pulled this problem from the 7th grade curriculum.

I like how it gets students thinking about the relationships to come up with true and false statements.  There is nothing that needs to be solved so students get to take it all in and notice relationships.  This was my entry point for what I wanted to accomplish.

I pulled some of the mobiles and removed any numbers at the top or in the solution so that students would not need to solve the mobile. Then we focused on making statements that must be true.  

My original plan was to slowly transition to algebraic notation.  I wrote one statement and I was ready to start short-handing it.  The students caught on very quick to how what they said translated into an equation.  This mobile for example, we know that 3 moons and a hexagon weigh the same as two triangles and a hexagon, 3m + h = 2t + h.  The simplicity of this mobile allowed us to push our thinking forward.  That is really the only statement that came up at first.  Then we noticed (or I nudged them to think about) that there is a hexagon on each side of the mobile.  For some students pulled out my balance scale and removed counters from each side so they could think about how removing items affects the scale.  We were able to come to the conclusion that 3m = 2t.  Our algebraic work from our discussion looked like this:

Each image of a mobile added another layer to our algebraic thinking and work. 

This mobile allowed us to remove drops from both sides, but also think about what balances with just one square.  If you do this you will want some way to draw all over your mobile, either on a Smartboard or with an annotating extension.  We marked up the mobiles a lot in our discussions.  

By this point I was officially asking for equations, although we were still speaking in shapes rather than variables.  This next mobile added some fun discussions.  

There are 2 parts to the mobile and many shapes that repeat.  There are also two equations with 3t on one side.  This allowed us to use our equations to create other equations (shown in green), which is actually used in solving systems of equations with substitution.   We had some discussions with whole number equations to help understand this.  3+4=7 and 5+2=7, so we know that 3+4=5+2.

This was such a good conversation that I added another mobile for my groups tomorrow. 

Here are the notes I've written for myself to help guide the discussion.

I was so impressed by my students ability to think about the number relationships in this way.  By focusing on making the logic explicit and connecting it to the algebraic notation, I was able to have some great conversations with students who have not solved equations before.  I am hoping when we get to solving equations more formally they have a deeper understanding of the relationships so that they don't feel like they are simply following rules.  

Here is the presentation I used.  It has some Open Up lessons as well as what I plan to do after this activity.  Depending on when you read this it may still be a working document.  I will blog about the rest in a separate post if it goes well.  

I would love to know your thoughts or if you try this activity.  Or if there is another way you connect the mobiles to algebraic reasoning.  Leave a comment below.