P E MD AS: aka Order of Operations

Please Excuse My Dear Aunt Sally.  There are a number of mnemonic ways to remember the order of operations, but I have never seen a hands-on demonstration that might help a child remember what to do first.  So here goes:

Quick review: if you don't recall, a math operation is the adding, subtracting, multiplying, and dividing that is applied to a set of numbers.  An expression is a combination of numbers and operations that can be simplified to a single value, in other words, whatever is on one side of the = in an equation.  When there are several operations to be done in an expression, mathematicians have agreed to do them in a certain order.   For example,  3 x 4 + (6+2) + 5 has a lot going on.  But just like in the emergency room, where the first patient in the door is not necessarily the one that gets seen first,  the numbers are not "operated on" from left to right. PEMDAS is the triage system that gets numbers in a long expression taken care of. 

So here is how to set it up:  The first thing you need are parentheses, and you'll make them out of paper.  Glue a pair of paper parentheses onto each of  few paper plates.  Now you need operations symbols, so glue some toothpicks or craft sticks together into addition (+) signs, which also double as multiplication (x) signs.  A dab of clay serves as the dot form (*) of the multiplication sign.  A single toothpick or stick serves as a subtraction (-) sign, and adding clay dabs above and below the stick turns (-) into division (÷).

Note:  the toothpicks are not glued onto the plates.

Either the clay dot or the toothpicks can be used as a multiplication sign, whichever your child is most familiar with.

Two clay dots turn a subtraction sign into a division sign.

The clay dots should be kept soft and sticky for a reason.  If you use clay to form numbers, it is best to use clay that can harden, in contrast to the "stickiness" of the multiplication and division symbols.



Now all you need are some 3-D numbers.  These can be magnetic numbers from the Dollar Store, foam puzzle numbers, numbers you form out of hardening clay or pipe cleaners (aka "chenille stems"), or in a pinch, you can cut numbers out of paper.

Use the numbers and operation signs to construct the expression.  If the numbers are in parentheses, they go on a plate.  For exponents, you can use a dab of clay-- aka the multiplication sign-- to stack numbers on top of each other, as many numbers as the exponent indicates.

So, for example, to indicate 4 to the third power, you would stack three 4's together with a dab of clay between each.   Use another piece of clay to form a small "3" to indicate the exponent.  For 5 squared, you would stack two 5's together with clay between them and a small clay "2" in the exponent position.  Note that the exponent itself is soft and sticky like the multiplication symbol, not hard like the numbers being multiplied. This model helps the student remember that 5 to the second power is actually 5x5 and not 5x2.

Now here is the explanation for the acronym "PEMDAS":

1. The P stands for "parentheses."  Or "plate." Either way, the parentheses form the outer shape of a plate, and you have to "clean your plate" first.  Or if you don't like that analogy, just stick to "do whatever is on the plate first."  So in this case, 6 + 2 is what you work out first.  6 + 2 = 8.

2. E is for Exponents.   Here we have 5^2, which is also know as 5x5, or 25.

Note: when typing exponents into a format that doesn't allow the little numbers in the exponent position, they are entered with a carat, like this:  5^2, which means "five to the power of two." Some computers have a calculator function that recognizes the carat also, and will calculate it for you.


Similarly, the asterisk * serves as a multiplication symbol, and the / forward slash as a division symbol.  If you want to see the ÷ symbol, try "alt" and /  or "option" and /.

3. Next, you look at the operation signs.  You will notice that both - and + have only a single horizontal line connecting the numbers on either side.  The vertical line on the + doesn't touch the numbers.  So these are kind of weak-looking links.  If you are using "x" as your multiplication sign, the "x" looks like a stronger link than addition, because it can touch both numbers in two places, top and bottom.  (Yes, I know that in real life the x doesn't actually touch the numbers, but we'll imagine.) The "dots" can be thought of as "sticky"-- like clay or glue dots or magnets or even planets with gravitational fields.  So they also make stronger ties than + or -.

So, after you take care of whatever is on the plates, and simplified the exponents, you go on to the strong/sticky operations.  That is the M and D, multiplication and division, of PEMDAS.  (Note that multiplication and division are done left to right, so it could just as easily be called PEDMAS.)

Last, you take care of the weak links, with addition and subtraction.  Again, these are done left to right, so if your - sign comes before your addition, you subtract first-- but PEDMSA is a lot harder to say.

So then,                       3 * 4 + (6+2) + 5^2:
Plate/parentheses:                       8
                                    3  * 4 +    8     + 5^2
Exponents                                               25
Sticky ops:                   3 * 4      
Weak links:                     12    +  8      +  25
Solution:                                              =  45


Easy, right?

Now,  if the exponent were on the outside of the parentheses/plate-- as in  3 x 4 + (6+2)^2 + 5^2, you would still do PEMDAS order.

In this case, you may need to help the student model the idea that since the whole "plate" is being squared, you can make two identical plates and stack them on top of each other.

Then it  obvious that you have to simplify the expression inside the parentheses-- add 6 +2 -- before you can square it.

So then,            3 * 4 + (6+2)^2 + 5^2:
Plate/parentheses:               8^2
                          3 * 4 +    8^2     + 5^2
Exponents                          64          25
Sticky ops:        3 * 4      
Weak links:         12    +  64      +  25
Solution:                                   =  101


If your student doesn't see the importance of "triaging" the numbers in the same order each time, you will of course want to take him through the operations the right way (PEMDAS) and the wrong way (straight across left-to-right or randomly) just to see the difference.  Note-- if the equation is already written so that multiplication or division happen first, it may not be as effective to compare the two results.  Try something like   3 + 4 * 6 + (4 - 2)^2  which if not done correctly, will not give the correct result of 31:

3 + 4 * 6 + (4 - 2)^2
3 + 4 * 6 + 2^2
3 + 4 * 6 + 4
3 + 24 + 4
31

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.