Today’s Objective: To understand and use properties to write and solve expressions and equations. Why is it important? Using properties makes it easier to work complicated problems.
Properties A property is something that is true for all situations.
Four Properties 1.Distributive 2.Commutative 3.Associative 4.Identity properties of one and zero
Look at this problem: 2(4 + 3) Through your knowledge of order of operations, you know what to do first to evaluate this expression. 2(7) 14 Now, look what happens when I do something different with the problem. 2(4 + 3) = 8 6 += 14 No difference. This is an example of the distributive property. **
Now why would one ever use the distributive property to solve 2(4 + 3)? The answer is generally, “You wouldn’t! Just use the order of operations.” One place this is going to become very important is when we have an expression in the parenthesis which can not be simplified, like: 2(4 + x) You need to be able to recognize and use the distributive property throughout all of Algebra. This is one property you need to know by name, forwards, and backwards!
EXAMPLE: Use the distributive property to find each product. a. 7 * 98b. 8(6.5) First break down (decompose) the number 98: 7(90 + 8) Then distribute Finally, add. 686 How can we decompose 6.5? Hint: How do we read the decimal? 8( ) Then distribute Finally, add. 52 The distributive property can make large calculations easier for using mental math.
Distributive Property A(B + C) = AB + AC 4(3 + 5) = 4(3) + 4(5)
Commutative Property of addition and multiplication Order doesn’t matter A x B = B x A A + B = B + A
Associative Property of multiplication and Addition Associative Property of multiplication (a · b) · c = a · (b · c) Example: (6 · 4) · 3 = 6 · (4 · 3) Associative Property of addition (a + b) + c = a + (b + c) Example: (6 + 4) + 3 = 6 + (4 + 3)
Identity Properties If you add 0 to any number, the number stays the same. A + 0 = A or = 5 If you multiply any number times 1, the number stays the same. A x 1 = A or 5 x 1 = 5