Unit 1 Relationships Between Quantities and Expressions

Week 1

Lesson 2

Standards/Objective• A.SSE.1 – Interpret parts of an expression• A.APR.1 – Add, subtract, and multiply

polynomials; understand that polynomials form a system analogous to the integers in that they are closed under these operations.

Objective: Students use the structure of an expression to identify ways to rewrite it. Students use the distributive property to prove equivalency of expressions.

Essential Questions

• Why are the Commutative, Associative, and Distributive Properties so important in math?

• What role does the Commutative Property play in proving equivalency of expressions?

• What role does the Associative Property play in proving equivalency of expressions?

• What role does the Distributive Property play in proving equivalency of expressions?

Vocabulary

• Simplify• Equivalent Expressions• The Commutative Property of Addition: If a and b are

real numbers, then a+b=b+a.• The Associative Property of Addition: If a, b, and c are

real numbers, then a+b+c=a+(b+c).• The Commutative Property of Multiplication: If a and b are real numbers, then a×b=b×a.

• The Associative Property of Multiplication: If a, b, and c are real numbers, then abc=a(bc).

Read, Write, Draw, Solve

Angie says that the equation below is showing that the two expressions are true because she applied the Distributive Property to generate the second expression. Do you agree or disagree? Explain your reasoning.

2x + 4 = 2(x + 2)

Activator

• Roma says the collecting like terms can be seen as an application of the distributive property. Is writing x + x = 2x and application of the distributive property?

What expression can Diagram A and B represent?

Diagram A

Diagram B

Are the two expressions equivalent?Which property is this?

What expression can Diagram A and B represent?

Diagram A

Diagram B

Are the two expressions equivalent?Which property is this?

Use graph paper to draw an area model that can represent the expression 3 x 4.

Use graph paper to draw an area model that can represent the expression 4 x 3.

Do both models have the same area? Why?What property do we call this?

Use cubes to create a model that can represent the expression (3 x 4) x 5.

Use cubes to create a model that can represent the expression (4 x 5) x 3

Did you use the same amount of cubes in both models?

What property do we call this?

Start: (5 + 3) + 7

A: 5 + (3 + 7)

B: 7 + (3 + 7)

What property(s) can you us to show that (2 + 1) + 4 = 4 + (1 + 2)

What property(s) can you us to show that (2*1)4 = 4(2*1)

What property(s) can you us to show that (x + y) + z = z + (x + y)

What property(s) can you us to show that (xy)z = z(xy)

2(x + 3) + x + 3x + 5 = 2x + 6 + x + 3x + 5 ______________________= 2x + x + 3x + 6 + 5 ______________________= (2x + x) + 3x + 11 ______________________= 3x + 3x + 11

______________________= 6x + 11

Final Question!! Is 2(x + 3) + x + 3x + 5 = 6x + 11? Why or why not.

We now know that Algebraic Equivalence is when two algebraic expressions are equivalent if we can convert one expression into another by repeatedly applying the Commutative, Associative and Distributive properties and properties of rational exponents to components of the first expression.

In other words! If we can rewrite and expression applying one or more of the properties then the two expressions are equal!!

Exit Ticket

A. How can you prove the algebraic equivalence of 5(x + 4) and 5x + 20. Explain using

today’s vocabulary.B. How can you prove that R + T is equivalent to

T + R for any real number?C. Is (2x + 3) + 5 equal to 2x + (3 + 5)? Explain

why or why not.