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. A polynomial is any expression that is a combination of more than 1 term or monomial and separated by an addition or subtraction sign. Polynomials are continuous and defined for all values. You can replace x with any REAL number and get a REAL number as a result. Link for lots of Technology resources

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) Discuss with students how if you apply the distributive property to generate an expression from the original expression that the two expressions will be true for all real numbers.

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? Roma is correct. The the distributive property we have x + x = 1(x) + 1(x) = (1+1)x = 2x. This exercise also helps students see that there is always and understood 1 in front of x.

What expression can Diagram A and B represent. Diagram A. Diagram B What expression can Diagram A and B represent? Diagram A Diagram B Are the two expressions equivalent? Which property is this? You can use snap cubes for a more concrete representations. You may want to do more examples to drive the property home. Associative Property of Addition

What expression can Diagram A and B represent. Diagram A. Diagram B What expression can Diagram A and B represent? Diagram A Diagram B Are the two expressions equivalent? Which property is this? (3 + 4) + 2 = (2 + 4) + 3 You can use snap cubes for a more concrete representations. You may want to do more examples to drive the property home. Commutative Property of Addition

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? You can use tiles for a more concrete representations. You may want to do more examples to drive the property home. Communative Property of multiplication

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? You can use cubes, snap cubes, a drawing, picture or technology to create the models. You may want to do more examples to drive the property home. Associative Property of multiplication

Start: (5 + 3) + 7 A: 5 + (3 + 7) B: 7 + (3 + 7) DIRECTIONS: Explain what property is done to change the starting expression to the next expression Application of assoc. property of add Application of comm. Property of add

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) Assoc. and Comm. Of addition Assoc. and Comm. Of multiplication

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) comm. Of addition Comm. of Multiplication

2(x + 3) + x + 3x + 5. = 2x + 6 + x + 3x + 5 ______________________ 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. Distributive, Commutative, Associative, Associative. Yes they are equalivent because of the properties.

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!! Distributive, Commutative, Associative, Associative. Yes they are equalivent because of the properties.

Exit Ticket How can you prove the algebraic equivalence of 5(x + 4) and 5x + 20. Explain using today’s vocabulary. How can you prove that R + T is equivalent to T + R for any real number? Is (2x + 3) + 5 equal to 2x + (3 + 5)? Explain why or why not.