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Balancing Algebraic Understanding

6-8
1
Algebra
Rhonda Naylor
Location: unknown

Using a balance in the classroom is a first step to algebraic understanding. Use this pan balance (numbers) applet to practice the order of operations in simplifying numerical expressions and to demonstrate the conventions of using algebraic logic in simplifying expressions.

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.

overhead Who is Correct? Overhead 

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.

appicon Pan Balance — Numbers Applet

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.

overhead Discover OoOPS! Overhead 

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.

pdficon Balancing Expressions Activity Sheet 

You may also distribute Balancing Exponents Activity Sheet as time permits.

pdficon Balancing Exponents Activity Sheet 

Assessment Options

  1. The Discover OoOPS! Overhead will be a quick and easy assessment tool which can be used at the end of the lesson.
  2. Provide problems such as 72 ÷ 9 × 2. Do students multiply first and incorrectly get 4 [rather than 16]? If so, provide additional instruction to show that multiplication and division are completed in left to right order.

Extensions 

  1. Many pan balances use weighted numbers, or distance and weight, to demonstrate these relationships. For example, 2 weights at 8 will balance with 4 weights at 4. Show the same problem on the pan balance on the computer, and the classroom balance. Students are now seeing distance × weight = distance × weight (depending on the balance you have). The left side must equal the right side to maintain equality. To use the equal sign, students must know that this means the left side equals the right side.
  2. Use the Pan Balance: Expressions to move on to balancing variable expressions.
  3. Move on to the next lesson, Algebra in Balance.
 

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.]  

Teacher Reflection 

  • 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?
 
Unit Icon
Algebra

Everything Balances Out in the End

6-8
Use a pan balance to study equality, order of operations, numerical and variable expressions, and other key algebraic concepts.
2144icon
Algebra

Algebra in Balance

6-8
Students use the Balance Pans Applet- Expressions Tool to explore algebraic expressions. They determine if algebraic expressions are equal. They balance pans to solve a system of equations and use graphing to find the solutions to a system of equations.
PanBalanceShapes
Algebra

Balancing Shapes

6-8
Students will balance shapes on the pan balance applet to study equality, essential to understanding algebra. Equivalent relationships will be recognized when the pans balance, demonstrating the properties of equality.

Learning Objectives

Students will:

  • Develop an understanding of equality.
  • Use a pan balance (number) to determine equivalence of numeric expressions involving the order of operations (including exponents).
  • Develop algebraic understanding as expressions are simplified and recorded step by step.

NCTM Standards and Expectations

  • Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.