“Without any doubt, the foundational skill of
algebra is fluency in the use of symbols.”
Final Report of the National Mathematics Advisory Panel
Report of the Task Group on Conceptual Knowledge and Skills, Page 17
The NCTM Principles and Standards for School Mathematics notes that “a strong foundation in algebra should be in place by the end of the eighth grade.” Hands-On Equations is the unique program that is able to provide students with that foundation beginning in the 3rd and 4th grades. One of the greatest stumbling blocks that students have to the learning of algebra is the abstract nature of the symbolism that is used. Indeed, the final report of the National Mathematics Advisory Panel noted (page 60) that, “Many students have difficulty grasping the syntax or structure of algebraic expressions.” This is another way of saying that for many students algebra is a foreign language.
Hands-On Equations performs the essential function of demystifying algebraic notation through its unique visual representation of equations using pawns and numbered cubes. Almost instantly students understand the elements or makeup of an equation such as 4x + 3 = 3x + 9. They understand the essential way, for example, in which the 3 constant on the left side differs from the 3 which is coefficient that is on the right side. Once students understand what an equation means, and once they understand a few basic principles made very clear through the use of physical actions or gestures, they attain a very high level of success with algebraic linear equations normally presented only in an algebra course. Students are then able to apply that learning to the algebraic solution of verbal problems.
In Level I, the first seven lessons of Hands-On Equations, students learn:
- the concept of an unknown
- the relational meaning of the equal sign (both sides have the same value)
- the meaning of an equation
- how to balance equations (using the subtraction property of equality)
- the concept of the check of an equation
- the ability to solve one and two-step equations with unknowns on both sides
- how to combine like terms
- how to work with a multiple of a parenthetical expression, i.e., the distributive property
- how to evaluate an expression (when they check each side of an equation)
In Levels II and III of Hands-On Equations, students learn:
- 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
Hands-On Equations is an Essential Component
of Middle School Mathematics
Dr. Borenson, the inventor of Hands-On Equations, strongly urges all districts to provide at a very minimum Level 1 of Hands-On Equations, which consists of the first seven lessons of the program, followed by several lessons where the students apply this learning to the algebraic solution of verbal problems -- before students enter an Algebra 1 course. This will be the best investment districts can make to help their students succeed with the abstract world of algebra!
"A strong foundation in algebra should be in place by the end of eighth grade...".
Principles and Standards for School Mathematics, NCTM
Hands-On Equations Objectives
The above link will provide the objectives for Hands-On Equations, Level I (the red manual), and for the Hands-On Equations Introductory Verbal Problems Workbook and the Hands-On Equations Verbal Problems Book, Level 1.
NCTM Math Standards Correlation
Common Core State Standards
GRADES 3 - 5
GRADES 6 - 8
Our two-day training will provide your teachers with the skills to help their students meet these Common Core algebra standards:
- Solve two-step word problems and represent those problems using equations with a letter standing for the unknown. (3.OA.8). Click here for video demo.
- Solve multistep word problems using drawing and equations with a symbol for the unknown (4.OA.2). Click here for video demo.
- Use visual models and equations to represent and solve word problems involving addition and subtraction of fractions (4.NF.3d). Click here for video demo.
- Apply the property of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2+x) to produce the equivalent expression 6+3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y) (6EE.3)
- Understand that positive and negative numbers have opposite values (6.NS.5)
- Solve mathematical problems using algebraic expressions and equations (7.EE.4). Click here for video demo.
- Recognize linear equations which have no solutions, one solution, and infinitely many solutions (8.EE.7)
- Strategically choose process for solving equations in one variable (grade 8)
IMPORTANT: The staff training we provide to your teachers will enable your students to excel algebraically and go beyond the Common Core standards. For example, we make it possible for third and fourth graders to meet the above 7th and 8th grade standard and solve word problems using algebraic equations. Not only will your students have the skill and knowledge to solve algebraic equations with unknowns on both sides, they will experience the joy of learning and the enhanced self-confidence that comes from real achievement. Click here to see what district leaders say about our training.