What is Hands-On Equations®?

Making Algebra Child’s Play®

Hands-On Equations and Making Algebra Child's Play are registered trademarks of Borenson and Associates, Inc.

What is “Hands-On Equations”?

Hands-On Equations® is a visual and concrete approach enabling students in grades 3 – 8 to solve math equations.  The system uses game pieces, namely pawns and numbered cubes, and an image of a balance scale, to enable students to represent and solve algebraic linear equations. Students develop a conceptual understanding of the meaning of an equation, the unknown, and balance, thereby building self-confidence and a strong foundation for later math work.  Developed by Dr. Henry Borenson, this teaching system was awarded a U.S. patent.

How it works:

  • Manipulatives: Students use pawns (representing ‘x’) and number cubes (representing integers) on a laminated balance scale.
  • Setting Up Equations: They physically place the pieces to represent the given equation (e.g., two pawns and a ‘5’ cube on one side, a ’13’ cube on the other for 2x + 5 = 13).
  • Legal Moves: Students learn “legal moves,” like taking away equal amounts from both sides or removing pairs of opposite pawns, to physically simplify the equation until the ‘x’ (pawn) is isolated.
  • Concrete-Pictorial-Abstract (C-P-A): It follows a C-P-A approach, moving from physical objects to drawings (pictorial) and finally to abstract numbers and symbols, solidifying understanding.

Key benefits:

  • No Prerequisites: Students don’t need prior algebra knowledge.
  • High Success Rate: The game-like, intuitive approach leads to early success, boosting confidence.
  • Versatile: Works with any math curriculum and is effective for average students, gifted students, and those with learning disabilities.

Watch how easy it is for a six-year-old child to solve this algebra equation.

Why should I teach Hands-On Equations to my students?

Hands-On Equations is an innovative program designed to make algebra accessible and enjoyable for students. These are among the reasons we recommend you teach the program to your students:

  • No algebraic prerequisites are required.
  • It is a game-like approach that fascinates students.
  • The gestures or “legal moves” used to solve the equations reinforce the concepts at a deep kinesthetic level.
  • The program can be used as early as the 3rd grade with gifted students, 4th grade with average students and 5th grade with LD students; it also serves as an excellent component of a middle-school pre-algebra program.
  • Students attain a high level of success with the program (see research studies section).
  • The program provides students with a strong foundation for later algebraic studies.
  • The concepts and skills presented are essential for success in an Algebra 1 class.

Algebra concepts your students will learn in Level I (the first seven lessons):

Hands-On Equations provides a comprehensive approach to teaching foundational algebra concepts, empowering students with the tools and understanding they need to succeed. Through this program, students will learn:

  • The concept of an unknown
  • How to evaluate an expression
  • How to combine like terms
  • The relational meaning of the equal sign (both sides have the same value)
  • The meaning of an algebra equation
  • How to balance algebra equations (using the subtraction property of equality)
  • The concept of the check of an equation
  • The ability to solve one and two-step algebra equations
  • Solving equations with unknowns on both sides
  • How to work with a multiple of a parenthetical expression

In Levels II and III, students learn key concepts and strategies for solving equations.

Hands-On Equations offers a clear and engaging way to help students build a deep understanding of mathematical properties and procedures. With this program, students will explore:

  • The concept of the opposite of an unknown
  • How to evaluate algebraic expressions involving x and (-x).
  • The additive property of inverses
  • The addition property of equality
  • The additive identity property
  • The concept that subtracting an entity gives the same result as adding its opposite
  • Addition and subtraction of integers

Students learn so much more.

Hands-On Equations can transform a student’s perspective and help them discover the joy of learning mathematics. Through this program, students will realize that:

  • Mathematics is a subject one can understand.
  • Mathematics can be learned without memorization.
  • They need not be intimidated by algebraic symbols.
  • They can enjoy doing mathematics.
  • They can communicate their mathematical reasoning to others.
  • They can use concrete materials to model abstract equations and word problems.
  • They can have success in one of the most “difficult” topics of mathematics.
  • They have far greater learning potential than they ever realized.

“A strong foundation in algebra should be in place by the end of eighth grade…”
Principles and Standards for School Mathematics, NCTM