(2 x 5) + (2 x 3) = 16 FACTORING EXPRESSIONS 5 5 3 3 Factoring is an important process in algebra which is used to simplify expressions, simplify fractions, and solve equations. Consider the following: Jerry spent $5 for lunch on Monday and again on Tuesday. He spent $3 for snack on each day as well. How much did Jerry spend in all for both days? 5 5 3 3 2 x 5 2 x 3 (2 times the cost of lunch) (2 times the cost of snack) (2 x 5) + (2 x 3) = 16

Consider the following: Jerry spent $5 for lunch on Monday and again on Tuesday. He spent $3 for snack on each day as well. How much did Jerry spend in all for both days? Does the model shown below represent the situation as well? 5 3 5 3 5 + 3 5 + 3 (cost of lunch and snack (cost of lunch and snack on Monday) on Tuesday) 2 x (5+ 3) = 16

(2 x 5) + (2 x 3) = 2 x (5+ 3) Here’s another view… Let = $1 for lunch We can look at both expressions to see that they are equivalent. (2 x 5) + (2 x 3) = 2 x (5+ 3) Here’s another view… Let = $1 for lunch Let = $1 for snack

(2 x 5) + (2 x 3) = 2 x (5+ 3) (5 x 3) + (5 x 4) = ___ x ( __ + __) Mathematically, this equation shows an expression that has been “factored.” (2 x 5) + (2 x 3) = 2 x (5+ 3) GCF times THE OTHER SUM There is a shared factor (GCF) of 2. Take it out and multiply it to the remaining sum. PRACTICE: (5 x 3) + (5 x 4) = ___ x ( __ + __)

a a b b WRITE AN EXPRESSION REPRESENTING THE TOTAL. _____ _____ _____ _____ The expression that represents the total is ___________

Can you rearrange the parts of this bar to represent the total in another way? Now write a new expression to represent the same total. a b a b

The Distributive Property

3g + 3f = ________________ 6x + 9y = ________________ Use the GCF and the Distributive Property to write equivalent expressions. “Factor.” 3g + 3f = ________________ 6x + 9y = ________________ 3c + 12c = ________________ 24b + 8 = _________________

7x + 7y = _____________________ 15g + 20h = ____________________ 18m + 42n = _____________________ 30a + 39b = ____________________ 55m + 11 = ____________________ 7 + 56y = ______________________

Are these expressions equal? How do you know? 6x + 21y and 3(2x + 7y)

Evaluate each expression to prove that these two expressions are equivalent. Let g = 6 5g + 7g g(5 + 7)

Evaluate each expression to prove that these two expressions are equivalent. Let x = 10 14x + 2 2(7x + 1)

Fill in the blanks with the numbers that will make the equation true. 4x + 12y = ___ (x + 3y) 35x + ___y = 5 (7x + 10y) ___x + 9y = 9 (2x + y) 32x + 8y = 8 (___x + y)

Use models to prove that 3(a + b) = 3a + 3b

Use the GCF and the Distributive Property to write equivalent expressions in factored form. 4d + 12e 18x + 30y 21a + 28y

Distributing Expressions The expression 2(a + b) tells us that we have 2 of the (a + b)’s. Create a tape diagram representing 2 groups of (a + b). a b a b

Show how your model would look if we grouped together the a’s and then grouped together the b’s. What expression can we write to represent the new diagram? a a b b

Using Area Models to Help Distribute 2 (x + y)

Using Area Models to Help Distribute 2 (3x + 4y)

Using Area Models to Help Distribute y (4x + 5)

Using Area Models to Help Distribute 3 (7d + 4e)