Hands-On Equations Fractions

A concrete introduction to fractional linear equations

This 3-hour workshop shows teachers a concrete approach to introducing fractional linear equations. The workshop is appropriate for teachers of the gifted (2-5), middle school math teachers, and teachers of high school students struggling with algebra. Prior experience with the regular Hands-On Equations program is helpful, but not essential.

The unique HOE Fractions approach

Students first solve for the unit fraction of the unknown, and then for the unknown itself. For example, if a fractional equation is simplified to 7/3 x = 28, students see that 1/3 x = 4, and hence x = 12. In this manner, students use a sense-making approach to solving fractional linear equations.

Algebraic principles used in solving the equations

In solving the fractional equations of Level I, students use the subtraction property of equality and the substitution principle. They transform fractions into equivalent fractions having a common denominator. In solving the fractional equations of Level II, students use the addition property of equality, the property of additive inverses and the additive identity property.

We use the coaching model in our workshops

As in all of the Borenson Math workshops, we use the coaching model in which workshop participants work at their seats and present at the front of the room in order to internalize the concepts presented at the workshop. 

Basic concrete solution strategies

Workshop participants learn how to concretely represent fractional linear equations on their laminated balance scale. As with the regular Hands-On Equations program, they then use "legal moces" to simplify and solve the equations. The check is conducted in the "original physical setup."trategies are presented to enable workshop participants to transfer their concrete learning experience to abstract solution methods

A half-day 3-hour workshop

This workshop takes the teachers through the full Hands-On Equations Fractions program, comprising 12 lessons in total, as well as the enrichment lessons. Level I deals with fractional equations having positive constants. Level II deals with fractional equations having negative constants.


Option 1: Workshop & student kit for each participant

The district provides the basic workshop in which participants learn the concrete strategies that may be used to solve fractional equations. Each participant receives a HOE Fractions Student Kit.   

Option 2: Workshop & HOE Fractions Learning System

The district provides The Hands-On Equations Fractions Learning System for each participant, at the reduced price of $22.50 (instead of $29.50). This option provides the teacher with the instructional manuals and the student worksheets. Click here to see a sample page from the instructional page and here to see a sample worksheet.

 Option 3: Workshop & Class sets of materials

The district provides a class set of Hands-On Equations Fractions for each particpant, thereby enabling immediate classroom implementation, with each student having their full set of manipulatives. Depending on the number of class sets purchased, the workshop honorarium fee may be reduced or canceled.

Comments from workshop participants 

"This concrete approach allows students to progress to the abstract. The simplistic math manipulatives are powerful!"

"I really enjoyed learning how to represent fractional equations in a concrete manner. Love this!!" 

"It was amazing how quickly I grew in my content knowledge."

"I see this program enhancing student critical thinking and problem solving skills."

To bring this workshop to your district: 

Please send your inquiry to or call 800-993-6284. We would appreciate your providing two or three available dates on which we may provide the training.