Solving for Eggs... er, X.

Solving simple 2-step equations

Abstract algebra concepts are being presented to students of increasingly younger ages.  This can be a problem for students who are still very concrete thinkers.   One of my frustrations in trying to teach my sixth graders how to solve a two-step algebra problem was that I was never satisfied with the given models.  The whole purpose of modeling the 2-step equation is to get them to understand why the operations must be done in a certain order,  but the available models seemed to just confuse the issue. So instead, I came up with the Egg Solution.

The Egg Solution shows the student why the math operations must be done in a certain order.  It also helps the student understand that x represents the number that is unknown.

With this model, the variable x is represented by a plastic egg, and the coefficient tells how many eggs, or "x"'s, there are.  X is one egg, 2x is two eggs, and so forth.  The constants are represented by beads, buttons, beans, candies, popcorn kernels or Legos-- whatever fits in the egg.  A piece of paper that is blank except for a large = in the center is also helpful in modeling the equation.

For example, the 2-step problem 2x + 4 = 14 is set up like this:

I place 2 plastic eggs (each containing 5 beads) plus 4 loose beads on the right side of the paper, and 14 beads on the left side.

I tell the student, "This is a math game called 'solving for x.'  X stands for the number of objects in each egg.  To win, you have to figure out how many objects are in each egg without opening the eggs."

"On the left side, there are beads in the eggs, and each egg has the same number of beads; there are also some loose beads.  The total number of beads on the right side is the same as those on the left."

Some students will be able to figure it out intuitively, but might not be able to explain their thought process.  Others might need a hint:  "What would happen if you got rid of the extra four beads?"

Of course, the student might need to be guided to understand that if the four extra beads are removed from the egg side, they must also be removed from the non-egg side to keep the sides equal.

Once the extra beads are gone, it may be obvious to the student (or again, he may need a clue) that if the total number of beads in the eggs is equal to the number of beads on the right, and if each egg contains the same number of beads, then the number of beads can be divided by the number of eggs to see how many beads would be in each egg.  The solution, then, is x = 5.

Then the student may open the eggs to see if he was correct.

Once the student has mastered how to determine the number of beads in each egg, it is time to convert the eggs and beads into a written equation.  Model this first for the student, being sure to note that the number of eggs (coefficient) is represented by the number in front of the x, and the other numbers represent known quantities of beads.

Show the student how to write the steps to the solution:

2x + 4      = 14      
2x + 4 - 4 = 14 -4   (get rid of the loose beads)
2x             =  10
2x / 2        =  10 / 2  (divide the beads on the right side by the number of eggs)
  x             =   5         (3 beads in each egg)

Set up other equations using different numbers of eggs and beads.  What you're aiming for is:

1. The student can "solve for eggs" (or x) by manipulating the beads and eggs.
2. The student can write the equation from the beads and eggs that you set up.
3. The student can set up the eggs and beads to model an equation he is given.
4. The student can write the steps as she solves the equation using the beads and eggs.
5. The student can solve for x, writing each step correctly, without the manipulatives.