Carolin’s Notation for Solving Equations

Note: This entry is a non-technology post. Carolin is a visiting student from Germany incarolin200 my 5th Year Maths class.

To solve a basic algebraic equation, we isolate the unknown to one side of the equation. In the case of  x  as the unknown, we say, “we get the x term by itself.”

The steps we go through in order to get  x  by itself often requires us to either (i) add or subtract the same term to/from BOTH sides of the equation or (ii) multiply or divide the same factor to BOTH sides of the equation. By performing the same operation to both sides of the equation, we keep the equation balanced.

As discussed in class, we DO NOT use some strange type of VOODOO maths logic that allows us to move a term from one side of an equation to the other side and then magically “change” its respective sign.

As an example of documenting every step involved in solving a specific linear equation, consider the worked out solution below:


The highlighted terms emphasize how I’ve worked to keep the equation balanced every step of the way.

Not unreasonably so,  many teachers and students find documenting the operations on both sides of the equation laborious – overkill to be precise.

The problem with not writing out the steps on each side, however, is that it can lead to silly mistakes and, if the original attempt to solve the equation doesn’t work out, the process of finding where you made your mistake can be made more difficult than if you left a clear trail of your steps taken.

In steps Carolin!

Carolin is a visiting student from Germany who is in my 5th year mathematics class. In the process of marking one of her tests, I noted she had a “different method” for documenting her steps. Below is how she would document the solution to the above linear equation:

Carolin's Notation for Solving Equations

Carolin's Notation for Solving Equations

Carolin documents each step she is  “about” to take by way of a single reference in the right-hand margin. So as to avoid confusing her next step with the actual equation, she separates her “next step” from the equation with a small vertical line. It is implied that the operation in the margins is to be performed on BOTH sides of the equation.

Carolin’s notation for solving equations provides us with an easy to understand, concise method for documenting our steps. Along with assisting us in re-tracing our steps (in case of an error), it also provides the person checking our work with clarity on how we worked through our problem.

carolin200Carolin tells me she learned the notation as part of her maths studies in Germany. I’ve begun using Carolin’s notation with my second year maths students and it has proved very helpful. Thank you Carolin and thank you Germany!

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