Distributive Property Unit 1 Number System

Common Core State Standards CCSS.Math.Content.7.NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

The Distributive Property Replace the x-value with a number and see if these two ways of finding area will result in the same answer.

Using Distributive Property for Mental Math The Distributive Property is also helpful when you do math mentally. When you need to find the product of two numbers, write one of the numbers as a sum or difference. Then use the Distributive Property to help you find the product mentally. 9(31) = 9(30 + 1) Rewrite 31 as a sum. = (9  30) + (9  1) = 270 + 9 Multiply. = 279 Add.

Example Write each product using the Distributive Property. Then simplify. 8(59) 8(59) = 8(60 – 1) Rewrite 59 as a difference. = (8  60) – (8  1) Distributive Property = 480 – 8 Multiply. = 472 Subtract.

Practice Write each product using the Distributive Property. Then simplify 1. 4(98) 2. 7(32) 392 224

Does not work in this case… a(b×c)

Key Things to Remember The number of terms inside the parenthesis is how many times you will distribute Terms are separated by + or - Always remember to apply your integer rules!

Example 2 [ ] 3 1 – 20 + ( –5 ) = – 3 20 ( ) 1 + –5 = – 3 60 + –15 ( ) Distributive property 1 + –5 = – 3 60 Multiply. + –15 ( ) = – 72 Add from left to right.

Example 2 b. (-3+x)2

Example 3 12(2+x-3)

Distributing a Negative *When distributing a negative number, always remember to multiply the negative number to EACH term inside the parenthesis. –5 ( x + 10 ) = x –5 + ) 10 ( Distributive property = x –5 + –50 ) ( Multiply. = x –5 – 50 Simplify.

*A negative sign outside the parenthesis should be treated as -1 and distributed like a negative number. – ( x + 8 ) = x –1 + ) 8 ( Distributive property = x –1 + –8 ) ( Multiply. = x –1 + –8 Simplify.

Practice –8 ( 5 + x ) = (5) –8 + ) x ( = –40 + –8 x ) ( = –40 + –8 x – Distributive property = –40 + –8 x ) ( Multiply. = –40 + –8 x Simplify. – ( x – 2 ) = x –1 – ) 2 ( Distributive property = x –1 – –2 ) ( Multiply. = x –1 + 2 Simplify.

Example 4 A replica of the Parthenon, a temple in ancient Greece, was built in Nashville, Tennessee, in 1897. The diagram below shows the approximate dimensions of two adjacent rooms inside the replica. You can find the total area of the rooms in two ways, as shown. ARCHITECTURE Two methods can be used to find the total area of the two rectangular rooms.

Example 4 continued Area 63(44) = + 63(98) METHOD 1 Find the area of each room, and then add the two areas. 2772 = + 6174 8946 square feet =

Example 4 continued METHOD 2 Find the total length, and then multiply by the common width. Area 63(44 = + 98) 63(142) = 8946 square feet = ANSWER The total area of the two rooms is 8946 square feet.

Guided Practice Use the distributive property to find the area of the rectangle. 1. ANSWER 340 ft 2 Evaluate the expression or write an equivalent expression. 2. –2 + 5 ) ( 12 ANSWER –34 3. –4 – –7 ) ( 10 ANSWER 68 4. 2 – w ) ( 8 ANSWER 2 – w 16 5. –8 + z ) ( 25 ANSWER –8 – z 200