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 algebraic process.
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 pairs.
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
= 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."
For example:
9 * (8 + 4) - 5 = 9 * (12) - 5
= 108 - 5
= 103
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.
