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Algebraic Thinking and Common Core Standards – Grant Zimmerman

By February 21, 2023March 26th, 2023No Comments

While listening to the outstanding presentations at the ASCD Common Core Summit held on November 8, 2011 at Grandover Resort in Greensboro, NC, I couldn’t help thinking about how classroom teachers will both advocate and implement the standards. I look for ways to act on ideas. Early introduction of algebraic thinking in the Common Core Standards is one of those ideas that should be developed in elementary school.

Common Core Standards do not define:

  • How teachers should teach
  • All that can or should be taught
  • The nature of advanced work beyond the core

To me, this means that teachers will continue to design innovative lessons to advance students’ thinking. So, I’m listening, thinking, writing my ideas in OneNote and remembering how I used Hands-On Equations® to go beyond the core standards and teach algebraic thinking to fifth grade students. ( Hands-On Equations® , developed by Dr. Henry Borenson.)

Students use the Hands-On Equations® white and blue pawns, red and green number cubes, and a balance scale to solve equations throughout each of the three levels. By tactilely manipulating the components, students solve problems such as 4X+3=3X+9 during Level 1: Lessons #1-#7.

Level II introduces the concept of negative variables through a new mathematical notation which consists of an x with a bar through it (called “star”). The beauty of star or the opposite of X, allows for the tactile movement of negative values along the balance scale. Star is represented by the white pawn. X is the blue pawn.

So, in the equation 2X= Star + 6, a blue pawn (X) is added to each side. Because (-X) + X equals zero, the two pawns (blue and white) can be removed from the balance scale. Magic! The equation now reads, 3X=6.

Level III introduces negative integers. Red cubes represent positive integers. Green cubes are negative integers.

Key Point

Teach your students to write and solve the tactilely manipulated solution in the same manner expected of them in algebra class. Extend their understanding by completing the thinking process. You will not find this instruction within the Hands-On Equations® materials. I always asked my students to finish the equation by writing out the solution as if they were in first year algebra class.

Students learn how to solve equations and, at the same time, are able to explain what they did and what their manipulations mean. These actions are key attributes to the Common Core Standards.


Read and find a way to collaboratively discuss the Common Core Standards.

View the many YouTube videos about Hands-On Equations®. You can also find many videos on their Facebook page.

Have your students make their own equation-solving videos for parent night.

Grant Zimmerman is the Senior Education Consultant with Knowledge Network Solutions —Leaders in Technology Integration in Schools. Grant is also on the faculty of the National Paideia Center at the University of North Carolina. He leads educators in Professional Development sessions on the Paideia Seminar, the Paideia Project, and Technology Integration.