Students often resist showing how they simplify 6 + 8 × 2 ÷ 4. They
may say, "I know it's 10." They may incorrectly say, "7." The applet
used in conjunction with this lesson will demonstrate algebraic logic,
and convince students that mathematicians, and the computer, logically
display one step at a time, underneath each other. This leads students
to success with algebra.
Use the scenario on the overhead, Who is Correct?, to demonstrate the importance of the order of operations when determining the number of tiles needed to remodel a bathroom.
Ask students, "Which person is able to determine the accurate cost? Show how you know."
[Both Doug and Andy will get the correct amount, $365, as long as Andy uses the order of operations].
Ask students, "What error did Andy make? How can you help him correct his work?"
[He didn’t follow the order of operations, and went
left to right, instead, when adding 10 + 7. This is a common student
error. The correct steps are as follows:
10 + 7 × 40 + 5 × 15 = 10 + 280 +5 × 15
10 + 7 ×40 + 5 × 15 = 10 + 280 + 75
10 + 7 × 40 + 5 × 15 = 290 + 75
10 + 7 × 40 + 5 × 15 = 365]
Andy needs to practice simplifying numerical expressions with the
following lesson using Pan Balance-Numbers. Students will see the step
by step process the computer uses to record each step in simplifying
expressions; they will recognize the importance of following this
To demonstrate the pan balance, and how it works, project Pan Balance - Numbers Applet.
You may demonstrate several examples so students see how to work the
balance before sending students to explore on computers alone or in
For example, place 25 in the left pan; the balance is
unbalanced. Place 25 in the right pan. As the arm balances, the
equation, 25 = 25, is shown under Balanced Equations. Ask students,
"What is another way to write 25?" Replace 25 with their suggestions in
one of the pans. [5×5, 50÷2, 5^2 etc.]. Also insert an incorrect
expression such as 5 + 5; the pans do not balance, and nothing is shown
under Balanced Equations. This applet requires * for multiplication , /
for division, and ^ for exponents.
Move on to a new problem by clicking Reset Balance. Place
5 + 2 * 3 on the left. Ask the class what will balance this expression.
Some students may make a conjecture that it = 21, thinking 5 + 2 = 7,
and 7 * 3 = 21. If so, place this incorrect response in the right pan.
It will not balance. Ask students, "Why didn’t it balance?"
[ The order of operations must be followed. One must multiply 2 * 3 before adding 5]
Place 5 + 6 in the right pan. It balances, and the equation,
5 + 2 * 3 = 5 + 6 is shown. Now replace 5 + 6 with an equivalent
expression, 11. The Balanced Equations now shows:
5 + 2 * 3 = 5 + 6
5 + 2 * 3 = 11.
Students are seeing algebraic logic, recording one step at a time,
and the importance of the left side of an equation being equivalent to
the right side of an equal sign.
Focus not on the final solution, which is shown above the pan,
but the steps of justification which demonstrate HOW that becomes the
solution. Reinforce that multiplication and division must be completed
in left to right order; then addition and subtraction. A common student
error, because of the acronym "PEMDAS," or Please Excuse My Dear Aunt
Sally, that multiplication is performed before division.
5 ÷5 × 8 = 1 × 8
not 5 ÷ 5 ×8 = 5 ÷ 40 = 1/8!
Place increasingly difficult examples for students to see, and begin
to record on their own paper as shown under Balanced Equations.
"Left side, leave it; right side, keep simplifying, as the applet records the steps."
9 * (8 + 4) - 5 = 9 * (12) - 5
= 108 - 5
If it balances you know it is correct.
Reset balance and show another example: 5^2 + 7 - 3 on left side. It
is common for a student to write 10 + 7 - 3 on the right side. When the
balance doesn't balance, discuss exponents. "What does 5^2 mean?"
[5 raised to the second power, or 5 used as a factor twice; 5 * 5 = 25].
Students may then correctly suggest 25 + 7 - 3.
As a final activity, project the Discover OoOPS! Overhead for students.
The solutions for this overhead are available here.
Once students understand how the balance works and how to record their work step by step, distribute the Balancing Expressions activity sheet for students to complete individually or in pairs at a computer.
You may also distribute Balancing Exponents activity sheet as time permits.
Questions for Students
1. Why is important for everyone to follow the same order in simplifying expressions?
[If we didn’t, we would come up with various answers for the same numerical expression.]
2. What order does your own calculator follow when you enter 3 + 2 × 5?
[Scientific calculators will follow the order of operations. If a student has a 4 function calculator that does not follow the order of operations, it is important for students to input the information in the correct order themselves.]
3. What is the correct order of operations?
[Parentheses and other grouping symbols, exponents, multiplication and division in left to right order; then addition and subtraction in left to right order.]
- Were the concepts presented too concretely? Too abstractly? How would you change this in the future?
- How did the technology increase the engagement of your students? How could this be enhanced?
- What worked with classroom behavior management? What didn’t work? What needs to be improved?