Transcript Document
Chapter 3 Whole Numbers Section 3.4 Properties of Whole-Number Operations Why Learn Whole-Number Properties? Understanding whole number properties are important for two main reasons: 1. At the early childhood level it helps reduce the number of arithmetic facts and make computations easier. For example, you only need to learn 6+8=14 you also know 8+6=14. 2. At the middle childhood level they support the learning for algebra concepts. They are what really form the “rules” of algebra. For example you know that when dealing with the expression x+6 it is the same as 6+x. The Associative Property An association is a group. The associative property has to do with how numbers or expressions are grouped. Associative Property of Addition: For any whole numbers x, y, z: (x + y) + z = x + (y + z) Associative Property of Multiplication: For any whole numbers x, y, z: (xy)z = x(yz) For example: (2 + 3) + 4 = 5 + 4 = 9 and 2 + (3 + 4) = 2 + 7 = 9 2(34) = 212 = 24 and (23)4 = 64 = 24 How does the associative property apply to the curriculum? Is the basis for doing certain types of algebra problems. Simplify the following algebraic expressions: (x + 13) + 26 apply the associative property for addition we get x + (13 + 26) we can use the addition fact 13 + 26 = 39 x + 39 4(7x) apply the associative property of multiplication we get (47)x we can use the basic multiplication fact 47 = 28 28x This enables us to combine the various parts of the expression for which the numbers are known. The Commutative Property To commute means to move. The commutative property allows you to move different terms in an expression. Both addition and multiplication operations have the commutative property. Commutative Property of Addition: For any whole numbers x and y: x + y = y + x Commutative property of Multiplication: For any whole numbers x and y: xy = yx Examples include the following: 4 + 9 = 13 and 9 + 4 = 13 35 = 15 and We use this property in simplifying the following algebraic expressions: (2 + x) + 17 apply the commutative property of addition (x + 2) + 17 apply the associative property of addition x + (2 + 17) x + 19 we can use the addition fact 2 + 17 = 19 53 = 15 The commutative property can be applied in simplifying this algebra expression: (7x)8 apply the commutative property of multiplication (x7)8 apply the associative property of multiplication x(78) we can use the basic multiplication fact 78 = 56 x56 apply the commutative property of multiplication 56x Identity Property Both addition and multiplication have a special number that when the operation is applied changes nothing. For addition it is 0 and for multiplication it is 1. The number 0 is called the identity element for addition and the number 1 is called the identity element for multiplication Identity Property for addition: For any whole number x: 0 + x = x and x + 0 = x Identity Property for multiplication: For any whole number x: 1x = x and x1 = x The identity properties are particularly useful when solving algebraic equations: x + 7 = 12 subtract 7 from both sides (x + 7) – 7 = 12 – 7 apply the associative property x + (7 – 7) = 12 – 7 use the subtraction facts 7 – 7 = 0 and 12 – 7 = 5 x+0=5 apply the identity property x=5 The Distributive Property The distributive property allows you to switch the order you do addition and subtraction with multiplication. We said before we use the parenthesis to show when you want to add or subtract first. This shows how they can be removed. The Distributive Property of Multiplication Over Addition: For all whole numbers x, y, z: x(y + z) = xy + xz The Distributive Property of Multiplication Over Subtraction: For all whole numbers x, y, z: x(y - z) = xy - xz (In the reverse direction this is sometimes called factoring) The distributive property is one of the properties that is used to simplify algebraic expressions quite often. It is also one that gives many students difficulty. What properties are being applied in simplifying the algebra expression below: 3(5 + 4x) + 13 apply the distributive property of mult over addition (35 + 3(4x)) + 13 use the basic multiplication fact 35 = 15 (15 + 3(4x)) + 13 apply the associative property of multiplication (15 + (34)x) + 13 use the basic multiplication fact 34 = 12 (15 + 12x) + 13 apply the commutative property of addition (12x + 15) + 13 apply the associative property of addition 12x + (15 + 13) use the addition fact 15 + 13 = 28 12x + 28