Hands-On Equations Fractions 

 A concrete introduction to fractional linear equations  

Hands-On Equations Fractions (HOE Fractions), developed by Dr. Henry Borenson, provides students with a concrete introduction to fractional linear equations. The program may be used with academically talented students in grades 2 - 5, average students in grades 5 - 8, and high school students struggling with algebra. Materials may be purchased here.

The Hands-On Equations Fractions Learning System

The Hands-On Equations Fractions Learning System is the complete program with one set of student manipulatives. The program consists of two manuals encompassing 12 lessons, a set of worksheets and an answer key, a flat laminated balance, a set of 8 number cubes and a set of 33 fraction blocks.

Materials needed for each student

Hands-On Equations Fractions Student Kit 

Each student needs the HOE Fractions Student Kit to represent and solve the equations concretely. This kit consists of a set of 8 numbered cubes, a flat laminated scale, and a set of 33 plastic fraction blocks.


This program consists of two levels encompassing a total of 12 lessons (click here for a sample lesson). As with the original Hands-On Equations program, students represent the fractional equation on a flat laminated scale using their game pieces, and then use "legal moves" to solve the equations. 

Uniique to HOE Fractions is that students solve first for the unit fraction of the unknown. Hence, if a fractional equation has been simplified to 7/3 y = 28, they first determine the value of the unit fraction 1/3 y, namely, 4. Hence y, consisting of 3 copies of the unit fraction 1/3 y, is 12. This methodology provides students with a sense-making approach to fractional linear equations.

Each level ends with an abstract solution approach which may be presented to younger students as an enrichment lesson. For older students, the abstract solution approach provides a transition to solving these equations without the manipulatives. Level II has fractional equations with negative constants.

Materials needed for each teacher

Hands-On Equations Fractions Class Set 

The class set of materials includes The Hands-On Equations Fractions Learning System, as well as a HOE Fractions student kit for each student. After providing the two instructional examples for a lesson, the teacher distributes the worksheet for that lesson. Each worksheet has 8 equations, 4 from the current lesson, and 4 from prior lessons (click here for a sample worksheet). The teacher answer key enables the teacher to quickly check student solutions.

It is also highly recommended that the teacher obtain the HOE Fractions for the Interactive Whiteboard (Promethean, Mimio, etc.). A teacher without an interactive whiteboard can manipulate the digital game pieces from the computer keyboard. (This product is also available for the SMART Board.)


Hands-On Equations Fractions for the Interactive Whiteboard  

Staff Development

A half-day Hands-On Equations Fractions Workshop is available to assist teachers in implementing the program. Please write to info@borenson.com or call 800-993-6284. 

"The students truly loved the program! One learner's parent came up to me and told me how excited her daughter was to be working algebra using fractions!  It was a real boost to self confidence for several learners, especially to those typically struggling learners." - 5th grade teacher