1. Ch 1 Sec 6: Slide #1 Columbus State Community College Chapter 1 Section 6 Multiplying Integers

2. Ch 1 Sec 6: Slide #2 Multiplying Integers 1.Use a raised dot or parentheses to express multiplication. 2.Multiply integers. 3.Identify properties of multiplication.

3. Ch 1 Sec 6: Slide #3 Arithmetic vs. Algebra Arithmetic 4 x 7 = 28 Factors Algebra 4 7 = 28 or 4 ( 7 ) = 28 or ( 4 ) ( 7 ) = 28 Product FactorsProductFactorsProductFactorsProduct

4. Ch 1 Sec 6: Slide #4 Expressing Multiplication in Algebra EXAMPLE 1 Expressing Multiplication in Algebra Rewrite each multiplication in three different ways, using a dot or parentheses. Also identify the factors and the product. (a)8 x 9 8 9 = 72 or 8 ( 9 ) = 72 or ( 8 ) ( 9 ) = 72 The factors are 8 and 9. The product is 72. (b)5 x 30 5 30 = 150 or 5 ( 30 ) = 150 or ( 5 ) ( 30 ) = 150 The factors are 5 and 30. The product is 150.

5. Ch 1 Sec 6: Slide #5 Using Parentheses NOTE Parentheses are used to show several different things in algebra. When we discussed the associative property of addition earlier in this chapter, we used parentheses as shown below. 7 + (–5 + 5) 7 + 0 7 Now we are using parentheses to indicate multiplication, as in 4 ( 7 ) or ( 4 )( 7 ).

6. Ch 1 Sec 6: Slide #6 Multiplying Two Integers If two factors have different signs, the product is negative. For example, – 9 4 = – 36 and 2 – 6 = – 12 If two factors have the same sign, the product is positive. For example, 5 8 = 40 and – 3 – 7 = 21

7. Ch 1 Sec 6: Slide #7 – 1 – 5 = 5 – 1 5 = ? – 1 5 = – 5 Confirming the Rules for Multiplying Two Numbers 4 5 = 20 3 5 = 15 2 5 = 10 1 5 = 5 0 5 = 0 20 – 5 = 15 15 – 5 = 10 10 – 5 = 5 5 – 5 = 0 0 – 5 = – 5 – 1 – 5 = ? 4 – 5 = – 20 3 – 5 = – 15 2 – 5 = – 10 1 – 5 = – 5 0 – 5 = 0 – 20 + 5 = – 15 – 15 + 5 = – 10 – 10 + 5 = – 5 – 5 + 5 = 0 0 + 5 = 5 The product of two numbers with different signs is negative. The product of two numbers with the same sign is positive.

8. Ch 1 Sec 6: Slide #8 Multiplying Two Integers EXAMPLE 2 Multiplying Two Integers Find each product. (a) – 5 6 = – 30 If two factors have different signs, the product is negative. (b) ( – 7) ( – 8) = 56 If two factors have the same sign, the product is positive. (c) 2 ( – 12) = – 24 If two factors have different signs, the product is negative.

9. Ch 1 Sec 6: Slide #9 Multiplying Several Integers EXAMPLE 3 Multiplying Several Integers Multiply. (a) – 2 ( 4 – 8 ) Parentheses tell you to multiply 4 – 8 first. Both factors have different signs, so the product is negative. – 2 ( – 32 ) 64 Now multiply – 2 – 32. Both factors have the same sign, so the product is positive.

10. Ch 1 Sec 6: Slide #10 Multiplying Several Integers EXAMPLE 3 Multiplying Several Integers Multiply. (b) – 5 – 3 – 6 There are no parentheses, so multiply – 5 – 3 first. Both factors have the same sign, so the product is positive. 15 – 6 – 90 Now multiply 15 – 6. The factors have different signs, so the product is negative.

11. Ch 1 Sec 6: Slide #11 CAUTION In Example 3(b) you may be tempted to think that the final product will be positive because all the factors have the same sign. Be careful to work with just two factors at a time and keep track of the sign at each step.

12. Ch 1 Sec 6: Slide #12 Calculator Tip – TI-30X IIS (–) Example 3(b): – 5 – 3 – 6 Calculator Tip You can use the negative sign key on your TI-30X IIS calculator for multiplication and division. (–) xx 536= – 90

13. Ch 1 Sec 6: Slide #13 + – Calculator Tip – TI-30Xa Calculator Tip You can use the change of sign key on your TI-30Xa calculator for multiplication and division. Example 3(b): – 5 – 3 – 6 + – x =5 36 x – 90

14. Ch 1 Sec 6: Slide #14 Multiplication Property of 0 Multiplying any number by 0 gives a product of 0. Some examples are shown below. – 1 0 = 0 ( 0 )( 6 ) = 0 5,928 0 = 0

15. Ch 1 Sec 6: Slide #15 Multiplication Property of 1 Multiplying any number by 1 leaves the number unchanged. Some examples are shown below. – 8 1 = – 8 ( 1 )( 27 ) = 27 44 1 = 44

16. Ch 1 Sec 6: Slide #16 Using Properties of Multiplication EXAMPLE 4 Using Properties of Multiplication Multiply. Then name the property illustrated by each example. (a) ( 0 ) ( – 5 ) Illustrates the multiplication property of 0. (b) 314 ( 1 ) Illustrates the multiplication property of 1. = 0 = 314

17. Ch 1 Sec 6: Slide #17 Commutative Property of Multiplication Changing the order of two factors does not change the product. For example, 4 5 = 5 4 and 2 – 7 = – 7 2

18. Ch 1 Sec 6: Slide #18 Associative Property of Multiplication Changing the grouping of two factors does not change the product. For example, 6 ( 5 2 ) = ( 6 5 ) 2 6 ( 10 ) 60 = 30 2 = 60

19. Ch 1 Sec 6: Slide #19 Using the Commutative and Associative Properties EXAMPLE 5 Using the Commutative and Associative Properties Show that the product is unchanged and name the property that is illustrated in each case. (a) 9 4 = 4 9 This example illustrates the commutative property of multiplication. 36 = 36 (b) ( 3 8 ) 2 = 3 ( 8 2 ) This example illustrates the associative property of multiplication. 24 2 = 3 16 48 = 48

20. Ch 1 Sec 6: Slide #20 Distributive Property Multiplication distributes over addition. An example is shown below. 4 ( 3 + 5 ) = 4 3 + 4 5 4 ( 8 ) 32 = 12 + 20 = 32 4 ( 3 + 5 ) = 4 3 + 4 5

21. Ch 1 Sec 6: Slide #21 Using the Distributive Property EXAMPLE 6 Using the Distributive Property Rewrite each product using the distributive property. Show that the result is unchanged. (a) 2 ( 9 + 3 ) 2 ( 12 ) 2 ( 9 + 3 ) = 2 9 + 2 3 24 = 18 + 6 = 24

22. Ch 1 Sec 6: Slide #22 Using the Distributive Property EXAMPLE 6 Using the Distributive Property Rewrite each product using the distributive property. Show that the result is unchanged. (b) – 3 ( – 8 + 1 ) – 3 ( – 7 ) – 3 ( – 8 + 1 ) = – 3 – 8 + – 3 1 21 = 24 + – 3 = 21

23. Ch 1 Sec 6: Slide #23 Multiplying Integers Chapter 1 Section 6 – End Written by John T. Wallace