February 28, 2012 *Review by Maria Miller, M.Sc., author of HomeschoolMath.net *

**Hands-On Equations** is a supplementary program that teaches students to solve linear equations using a **balance scale** and physical **"pawns" and number cubes.** It is basically a program using a physical manipulative for modeling and solving linear equations—great for kinesthetic and visual learners.

Before this review, I had heard about it, and I knew it involved solving equations with a balance, as if "playing." However, once I saw the complete program, I was impressed—because the system of Hands-On Equations includes not only simple equations, but also the usage of **distributive property** (level I), **negative unknowns** (level II), **negative numbers** (level III), AND even word problems similar to typical word problems in pre-algebra or algebra 1.

So it is much more than just a system of modeling the simplest equations. In the end of level III, students are solving equations such as 2(−x + 6) − (−2) = 3x + 1! However, the system is beneficial EVEN if your child/student gets to only do level I, which consists of only 7 lessons, because in the end of Level I, students already solve equations such as 2(x + 1) + 5 = 3x + 6.

Basically, the program starts out with the simplest of linear equations, such as 2x = 6 or x + 5 = 10. Students use a blue **pawn for the unknown** (for the 'x') and **number cubes for the constants**. From level II on, they usea white pawn for (-x), also called a "star", and denoted with ~~x~~. The equations are modeled on the balance so that the left side of the equation goes on the left side of the balance, and similarly for the right side.

Students are instructed about "legal moves" with which to "play" with theequations until they arrive to the solution. The legal moves of course correspond to the regular principles used in algebra. For example:

- In level I, lesson 3, students are told they can remove the same number of pawns from both sides, and the scale will still balance.
- In level II, lesson 9, students are instructed that a pawn and a "star" (the white pawn) are opposites, canceling each other. (Pawn corresponds to x, and star to -x).
- In level II, lesson 12, students are taught about a "convenient zero"—essentially adding (x + (-x)) to one of the sides, after which it is possible to remove pawns or stars from both sides, whichever the need might be.

Once students have mastered the setup and solved the equations using the game pieces, they are also taught how to set up and solve the equations using paper and pencil and a pictorial notation. From this, they can then easily transition to regular algebraic notation using "x".

I started both my girls with this program. The older is 9, and is currently doing 6th grade in Math Mammoth. She's breezed through Hands-On Equations and it has been fairly easy for her. Setting up an equation for the word problems (in the Introductory Word Problems book) has been a little bit difficult at times. However, she can easily solve all the word problems in that book without using the balance scale.

The easiness may be partly due to the fact that I have included simple equations in Math Mammoth grades 4-6, and even one lesson in both Math Mammoth 4th and Math Mammoth 5th grade curricula about solving equations using a balance. Those two lessons correspond to the first few lessons of Hands-On Equations in difficulty.

I think she is now extremely well prepared to learn linear equations in prealgebra... we should go through that with great ease.

The younger also started out enthusiastically, but somewhere along the way the equations got too difficult and confusing to her—which is not surprising as she's only 7. But she had no difficulties in level I.

I think this is a **terrific supplement** to any math curriculum and truly helps students to **think algebraically** and teaches them **how to solve equations**. It is a great program for students in grades 4-6, preparing them for pre-algebra. You could ALSO use it with students who might lack motivation and who need to get that "spark" back, or just to show students how fun math can be!

Then, as far as prealgebrand and algebra students, being a physical manipulative, I feel it benefits MOST those students who have trouble understanding the abstract concepts in beginning algebra. It lets students TOUCH and FEEL and SEE and manipulate the pieces, which makes the concepts "come alive." I don't think it is necessary for strong algebra students—they can grasp the principles and "visualize" the equations even without a manipulative.

You might wonder, does this system REPLACE the instruction for algebra? No, not by any means. First of all, a typical algebra course (such as algebra I) covers many more topics than just linear equations. Secondly, Hands-On Equations does not deal with all possible variants of linear equations. It does not include cases where you have fractions or decimals in the equation.

However, it CAN replace the beginning instruction for linear equations (as in pre-algebra or algebra 1). After or during the program, students will need to learn the regular notation for solving equations so that they can then go on in their algebra studies to other topics and to linear equations with fractions and decimals.

If you are interested, check out the website and also their YouTube channel.

Hands-On Equations®. Price: $34.95 - a complete program for one student. You can purchase separately an instructional DVD, Verbal Problems Introductory Workbook, and Verbal Problems Book.