Students can often rattle off the acronym PEMDAS or "Please Excuse My Dear Aunt Sally" as being associated with the order of operations.
Putting this memory into practice can be more of a challenge. By
practicing the correct order with a motivating game of Bingo, students
will be more eager to be accurate in their calculations.
One misconception by students is that all multiplication should happen before all division because the multiplication comes before division
in the acronym. In fact, multiplication and division have the same
precedence and should be evaluated as they appear from left to right.
| Incorrect | | Correct |
Similarly, addition comes before subtraction in the acronym, yet they have the same precedence.
| Incorrect | | Correct |
Try giving students an additional example before starting the game.
Playing Order of Operations Bingo
To prepare the materials for the game, you will need to print the Order of Ops Bingo Sheet.
The first two pages contain 50 expression strips, which you will need
to cut out and place in a bowl, jar, or hat. The third page contains
two bingo cards; you will need to photocopy this sheet, cut the copies
in half, and distribute a sheet to each student.
The object of the game is to get five numbers in a row,
vertically, horizontally, or diagonally, just as in the regular game of
bingo.
NOTE: The operations used for this lesson are addition, subtraction,
multiplication, and division. None of the expressions contain exponents
or parentheses.
Distribute a Bingo card to each student before starting the game. Give students the following instructions:
- Choose one space on the board as the "free" space and write the word FREE.
- Choose numbers to write into the other 24 boxes on your Bingo
card. Make sure you choose numbers in the ranges given at the top of
each column. That is, numbers in the first column ("B") must be in the
range 1‑10, numbers in the second column ("I") must be in the
range 11‑20, and so on. [This ensures better distribution of the
numbers.]
- You are not allowed to repeat any numbers.
Place all of the expression strips in a bowl, jar, or hat, and
choose them one at a time. After each selection, write the expression
on the board or overhead so students can evaluate it. Students should
copy down and evaluate the expression on their own paper. For the first
few turns, you may want to model how the numerical value is determined
for the expression by writing in any applicable parentheses and going
through the steps of evaluation. Make sure you write out the steps,
just as you'd like to see the students do themselves. Once the number
is determined, students can look for the number on their Bingo card and
mark it with a pencil or a chip.
The value (i.e., the "answer") for each expression follows the expression on each strip, so be sure to share only the expression, saving the answer to verify a winner.
Keep picking expressions. Students should calculate the value
for each expression, and then mark the square with that number on their
card (if that number appears on their card, of course). When a student
believes that she has correctly completed a column, row or diagonal on
her card, she should yell, "Bingo!"
When the game has a potential winner, ask the student to call
out the numbers that make the winning row, column, or diagonal. With
the class, determine if the numbers that the winning student calls are
indeed values from expressions that have been called out to check the
math and verify the win.
To extend the game for another winner, change the rules to
require 2 runs of 5 chips, or framing the exterior square of the board
(16 pieces).
If students use chips instead of crossing off numbers with a
pen or pencil, then they can exchange cards and play again. In order to
start a second or subsequent game, all expressions used in the previous
game are returned to the bowl, jar, or hat for a fresh start.
Questions for Students
1. The order of operations says to multiply and divide first. What does this mean?
[It means that multiplication and division are performed before addition or subtraction. However, it does not mean that multiplication should be done before division. Multiplication and division have the same precedence, so if either multiplication or division occur with in an expression, perform these operations from left to right.]
2. What does it mean for addition and subtraction to have the same precedence?
[It means that addition should not be done before subtraction. If either addition or subtraction occur with in an expression, perform these operations from left to right.]
3. In the expression 3 + 4 × 5 – (3 + 2), explain the order in which the operations should be performed, and evaluate the expression.
[Operations within parentheses are done first, so add 3 + 2 = 5. This changes the expression to 3 + 4 × 5 – 5. Then, multiplication (and division, too, though there's none in this expression) are performed before addition and subtraction, so multiply 4 × 5 = 20. The expression is now reduced to 3 + 20 – 5. Finally, perform the addition and subtraction left to right to give 18.]
Teacher Reflection
- What are the most common misconceptions students have regarding the
order of operations? What can be done to break those misconceptions?
- What examples were most helpful in getting students to
understand the order of operations? What other examples would help
students to better understand the order of operations?