An Expression Has Infinitely Many Equivalent Expressions

Math Lesson: Equivalent Expressions / Grade Level: 6
Lesson Summary:
Students review properties of arithmetic operations and then relate concepts of balance to algebraic equations. They apply arithmetic properties to simplify algebraic expressions. Advanced students are challenged to create equations that represent each property. Struggling students reason through simplifying an algebraic expression.
Lesson Objectives:
The students will know…

an expression has infinitely many equivalent expressions.

The students will be able to…

apply the properties of operations to generate equivalent expressions.
identify two expressions are equivalent.
simplify expressions.

Learning Styles Targeted:
Visual / Auditory / Kinesthetic/Tactile
Pre-Assessment:
1)Write the following equations on the board and then ask students to explain the properties of addition that each represents:

3 + 2 = 2 + 3 and 3 × 2 = 2 × 3 (Commutative property)
(3 + 2) + 4 = 2 + (3 + 4) and (3  2) × 4 = 2 × (3 × 4) (Associative Property)
4(12 + 3) = 4(12) + 4(3) (Distributive Property)
3 – 2 = 1 and 2 + 1 = 3 (Inverse Operations of Addition and Subtraction)
6 ÷ 2 = 3 and 3 × 2 = 6 (Inverse Operations of Multiplication and Division)
3 × 1 = 3 and 3 ÷1 = 3 (Identity Property)
3 + 0 = 0 and 3 – 0 = 3 (Zero Property of Addition and Subtraction)
3 × 0 = 0 (Zero Property of Multiplication)

2)Leave the properties on the board for reference throughout the activities.
3)Identify students with little awareness of the properties of arithmetic.
Whole-Class Instruction
Materials Needed: notebook, pens, and pencils
Procedure:
Presentation
1)Give students 1 minute to conceive of a simple way to demonstrate balance and equality. Then have them demonstrate their creation to the class. It might be to balance a pencil on a narrow rim, or balance on one foot.
2)Then write the equation ab = ba and ask how the equation represents balance and equality. Explain that when the symbol = appears in an equation, both expressions in the equation, if correct, balance because they are equal in value.
3)Refer students to the properties of arithmetic from the Pre-Assessment activity. Explain that those properties are actually algebraic laws that they have been using when they add, subtract, multiply, and divide. Those same properties apply in algebra.
4)Students write the algebraic formulas for each of the properties by substituting the numbers for corresponding variables. Then students dictate the equations to you as you write them on the board next to the arithmetic properties.
5)Explain that these properties helpsimplify algebraic expressions and solve equations.

Write an equation for the commutative property 323 × 72 = 72 × n. Based on the commutative property, what is n? (323)
Have students suggest an equation to demonstrate the distributive property

such as 360 ÷ 24 = (240 ÷ 24) + (120 ÷ n). Based on the distributive property, what is n? (24)
6)Write the expression 3n + 3n + 3n. Ask how it could be simplified (9n). Confirm that students understand the 3n + 3n + 3n = 9n, that these are equal expressions.
Guided Practice
7)Write the following expressions on the board and have students reason through combining terms to simplify each expression.

17x + 39x (56x)
15x – x – 7x (7x)
3x + y + x + y (4x + 2y)
5a + 6b – a + b – 4a (7b)
(7 – 4)ab (3ab)

8)Confirm that only like terms can be combined and that the coefficients can be combined without affecting the value of the variables.
Independent Practice
9)Divide the class into pairs. Give students five minutes to do the independent practice activity. One student writes an algebraic expression (for example, 3x + 5x or 7xy + 3x – 2xy + 9x) and the other student simplifies it. Students take turns writing and simplifying. Remind studentsthey can only combine or simplify like terms.
10)Students in each group present one expression and how it was simplified. Students defend their reasoning.
Closing Activity
11)Refer to the Pre-Assessment activity and ask students how the properties of arithmetic are similar to and different from the properties of algebra.
Advanced Learner
Simplifying Expressions Using Properties
Materials Needed: notebook, pens and pencils
Procedure:
1)Challenge students to create equations with variables and coefficients that demonstrate each of the properties of algebra covered in the lesson.
2)Review results and allow students to explain their reasoning.
Struggling Learner
Translate into Symbols
Materials Needed:notebook, pens, and pencils
Procedure:
1)Pose this expression: 9(2r + 8t) + 4(2r – 8t).
2)Students use the distributive property to decompose it and the associative property of addition to simplify it. [(9x2r + 9x8t) + (4 x2r – 4 × 8t) = (18r + 72t) + (8r – 32t) = 18r + 8r + 72t-32t= 24r + 40t]
3)Review results and allow students to explain their reasoning.