Combining Like Terms and the Distributive Property Todays

Combining Like Terms and the Distributive Property

Today’s Learning Goal n n We will begin to learn how to combine like terms to simplify algebraic expressions. We will review the distributive property and use it to simplify algebraic expressions.

Definitions n n n In our past work with algebraic expressions, we saw parts to equations like the following: -x -1 x = 2 x -5 -3 y -14 6 6 The individual parts above are called terms. A term is a number, a variable, or a product/quotient of numbers and variables. Like terms are terms that contain the same variables (numbers are considered like terms).

Combining Like Terms n For the following picture, we could say that it is equal to 2 x. x x x 2 x 3 x y y y 5 y n What would you say is equal to the second picture? n What would you say is equal to the third picture?

Combining Like Terms n If we wanted to add these together, what would be a simpler expression for 2 x + 3 x + 5 y? Great… 5 x + 5 y! x x x y y y +5 x 3 x + 5 y So, we can combine like terms to simplify an expression. 2 x n n Terms with x-variables and terms with y-variables are not like terms so we cannot combine them!

Combining Like Terms n a) Simplify the following algebraic expressions: 3 x + 4 x – 5 – 2 x b) = 5 x – 5 c) 20 x – 12 + 14 – 12 x = 8 x + 2 n – 6 + 12 x – 3 x – 8 = 9 x – 14 d) -x + 5 – 4 x + 2 – 6 x – 8 y = -11 x – 8 y + 7 Remember that the sign in front of the number goes with that number!

Distributive Property n n Below is a large rectangular public pool. It is separated into two parts because one side is the shallow end and one side is the deep end. What is the area of the deep end (dark blue section)? Nice… 20*30 = 600 ft 2. 20 ft 600 ft 2 30 ft 9 ft

Distributive Property n n What is the area of the shallow end? Nice… 20*9 = 180 ft 2. What is the area of the whole pool knowing that the deep end is 600 ft 2 and the shallow end is 180 ft 2? Awesome… 600 + 180 = 780 ft 2. 20 ft 600 ft 2 180 ft 2 30 ft 9 ft

Distributive Property n n We could have determined the area of the whole pool by looking at the length and the width. How could we find the area of the whole pool using the length and width? Great… 20*39 = 780 ft 2. 20 ft Width = ft 2 2 600 ft 780 30 ft + 180 ft 2 9 ft = 39 ft

Distributive Property n n By splitting the pool in this way, we saw that the area of the whole pool was given in two ways. 20*(30 + 9) = 20*30 + 20*9 If both of these ways gave us the area of the pool, what must be true about these two expressions? 20 ft 600 ft 2 180 ft 2 30 ft 9 ft Excellent…the two expressions must be equal!

Distributive Property n The equivalence of the previous two expressions illustrates the distributive property of multiplication. 20 20*(30 + 9) = 20*30 + 20*9 n n 30 9 The 20 was distributed to the 30 and the 9 to get 20*30 + 20*9. The rectangular area of the pool was a way to illustrate why the distributive property works.

Distributive Property n n Below is the same rectangle, but it is split up in a different way. What is an expression that could represent the area of the entire rectangle using length * width? Super… 20 (15 + 3 + 3). 20 ft 15 ft 3 ft

Distributive Property n What is another expression that could represent the area of the entire rectangle by showing what you do to determine the area of each part? Fantastic… 20*15 + 20*3. 20 ft 20*15 ? 20 ? *3 15 ft 3 ft 20 ? *3

20 Distributive Property n 15 15 333 Again, the distributive property can be used to show the equivalence between these two expressions. 20*(15 + 3 + 3) = 20*15+20*15 +20*3+20*3 n The 20 was distributed to each of the terms inside the parentheses.

Distributive Property 5 n n n 5 x ? 20 ? x 4 A similar picture can be drawn to illustrate the distributive property with algebraic expressions. Consider the following picture above. What is an algebraic expression we could write for the area of the entire rectangle if we did length*width? Awesome… 5(x + 4). n What is an algebraic expression for the area of the entire rectangle if we showed the area of each piece? Excellent… 5 x + 20.

Distributive Property n 5 x The distributive property was used to get the equivalence between the two expressions. 5(x + 4) = 5 x + 20 4

Distributive Property n n -2 3 x 4 y 3 Let’s look at another example: What are the two algebraic expressions that are equivalent for this picture (notice the -2)? Tremendous…-2(3 x + 4 y + 3) = -6 x – 8 y – 6. n Numbers are multiplied to the numbers and the sign in front of the number goes with that number.

Distributive Property n Simplify each of the following expressions: a) -3(3 x – 4) = -9 x + 12 b) -5(6 x + y – 2) = -30 x – 5 y + 10 c) -1(-2 x – 5 y + 3) = 2 x + 5 y – 3 –*+=– -x = -1 x –*–=+ -() = -1()

4 – 2(3 x + 5) A Common Mistake n Consider the problem above. It was solved by two different students as shown below. Jalen’s Work Tobias’ Work 4 – 2(3 x + 5) = 4 – 6 x – 10 = 6 x + 10 = – 6 x – 6 Jalen did the subtraction first and then multiplied. Tobias did the multiplication first and then combined. Which is correct and why? Nice…Tobias is correct because of the order of operations! n

Partner Work n You have 20 minutes to work on the following questions with your partner.

For those that finish early a) 2 x 3( + 4) For the following figures, write an expression for the perimeter of the polygon. Then, simplify the expression. 9 9) + x ( 4 - 3 x + b) 2 x – 3 x+ 20 – 2 x 5 x (-3 x 1 2 x + – 1) 2(5 – x) 8 x 12 x 3 x(5 + 2) n -3(-x – 1) x 3 6( ) 1 –

Big Ideas from Today’s Lesson n Combine like terms to simplify an algebraic expression. The term on the outside of the parentheses must be multiplied to every term on the inside of the parentheses. You must complete multiplication before addition or subtraction according to the order of operations!

Homework n Pgs. 30 – 31 (15 – 27 odd, 56, 57)