Yes or No? Stay or Go?



Hands-On Equation Balance Beam
www.borenson.com

My Pre-Algebra college students have just started solving equations containing one unknown.  As I tell them, we have now become snoopy detectives looking for the unknown.  Their greatest difficulty is deciding what stays and what goes in an equation. In other words, which term should be cleared by using the inverse operation?  I always start this chapter using Hands-On Equations®.  I have used them for years, and it does give a visual for those concrete learners.  I also refer to the written equation as a teeter-totter or see-saw which must always stay balanced. (Notice that Hands-On Equations® uses a balance beam.)  We also discuss the commandment of "Whatsoever thou doest to one side of the equation, we must doest to the other" and its importance. (I admit that I was with Moses when he received the Ten Commandments, but it "fell upon me" to convey The First Commandment of Solving Equations to future mathematicians.

One Unknown

After much practice with the Hands-On Equations®, we move to actual written equations such as:  x + 9 = 12.  Here's the rub; a few of my students know the answer and do not want to show any of their work.  Maybe some of you have this type of student as well.  Since, after 30+ years, I am still unable to grade what is in their minds, I insist that all steps be written down.  I explain that it's like riding a tricycle to ride a bicycle to ride a unicycle.

First, I instruct the students to look at the equation and determine which terms are out of place.  (Side note: Because my students are easily confused, at the present, we keep all of the unknowns on the left side and all of the numbers on the right side of the equal sign.)  Let's go back to our sample of  x + 9 = 12.  Because the x is already on the left side of the equation, the students write a "Y" over it for "yes".  The 9 is on the wrong side of the equal sign, so the students write a "N" over it for "no". Finally, they write a "Y" over the 12 since it is the correct place.  The students know they must use the inverse operation of addition to clear the 9 because it is a "no".  They therefore subtract 9 from each side of the equation resulting in an answer of 3.

This may seem laborious to some, but what if the equation is:  3 = y - 4?  This always freaks my students out; yet, if they do the yes/no procedure, they will discover that they have one "yes" and two "no's" which means they can rewrite the equation as y - 4 = 3.  The problem can now easily be solved since it is a yes = no, yes problem.




Unknown on both sides
of the equation

The next step is what to do when an unknown appears on both sides of the equal sign.  Usually, my students are sure they are incapable of solving such a difficult problem, but let's use the yes/no method and see what it looks like.  Notice in the sample on the left that we have a yes, no = no, yes.  We start by clearing the no on the left hand side of the equation by using the inverse of -9.  We then go to the right side and clear the no of  y by using the inverse operation of addition.  (Yes, I am aware both can be cleared at the same time, but again simple and methodical is what is best for mathphobics.)  We then divide each side by 4 resulting in the answer of 3.  When the problem is completed, my students are amazed and proud that they could solve such a long equation.  (You might notice in the illustration, a dotted line is drawn vertically where the equal sign is.  This helps my visual students to separate the two sides of the equation.)

If any of you try this approach with your students or have a different method, I would love to hear from you.  Just leave a comment and a short statement of how this process worked for you or what process you use that is even better.  That way, we can learn from each other.

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Hands-On Equations® is Algebra for the visual and kinesthetic learner.  This system, developed by Dr. Henry Borenson, enables students (even those in 4th or 5th grade) to easily and enjoyably learn essential Algebraic concepts and skills. Dr. Borenson received a U.S. patent for his teaching invention.

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