1. Bell Work: The radius of a circle is 5cm. What is the area of the circle?

2. Answer: = (3.14)(5cm) = 78.5cm 2 2

3. Lesson 9: Rules for Multiplication of Signed Numbers, Inverse Operations, Rules of Division of Signed Numbers

4. The sum of three 2s is 6. Also the sum of two 3s is 6. 2 + 2 + 2 = 6 3 + 3 = 6 We get the same answer when we multiply. 3 x 2 = 62 x 3 = 6

5. We get the same result because multiplication is just repeated addition of the same numeral. Because of this, we can use the number line to explain the multiplication of 3 and 2 in two ways.

7. Now let us find the product of 2 and -3 on a number line. We can obtain the same answer by adding two -3s.

8. Thus we see that (-3) + (-3) = -6 So, 2 x (-3) = -6

9. But now if we attempt to use the number line to show -3 times 2 by trying to draw -3 arrows that are +2 units long, we find that the task is impossible. We do not know how to draw -3 arrows because any number of arrows we draw will be a number greater than or equal to 1.

10. Because it is impossible to use the number line to display certain multiplication problems as well as division problems, then we will generalize 3 rules for multiplication of signed numbers.

11. 1.The product of two positive real numbers is a positive real number whose absolute value is the product of the absolute values of the two numbers. (+3)(+4) = 12 2(9) = 184 x 5 = 20

12. 2. The product of two signed real numbers that have opposite signs is a negative real number whose absolute value is the product of the absolute values of the two numbers. (-2)(4) = -86(-2) = -12(-3)(5) = -15

13. 3. The product of two negative real numbers is a positive real number whose absolute value is the product of the absolute values of the two numbers. -2(-3) = 6(-5)(-3) = 15-4(-5) = 20

14. Inverse Operations*: When one operation undoes another operation.

15. 7 + 3 – 3 = 7 7 + 3 – 3 = 7 Thus addition and subtraction are inverse operations. 7 x 2 = 7 7 x 2 = 7 2 Thus multiplication and division are inverse operations also.

16. Rules for division of signed numbers: 1.If one positive real number is divided by another positive real number, the quotient is a positive real number whose absolute value is the quotient of the absolute values of the original numbers. 6/3 = 28/2 = 412/4 = 3

17. 2. If a negative real number is divided by a positive real number, the quotient is a negative real number whose absolute value is the quotient of the absolute values of the original number. -6/2 = -3 -10/5 = -2-12/3 = -4 -6/2 = -3 -10/5 = -2-12/3 = -4

18. 3. If a positive real number is divided by a negative real number, the quotient is a negative real number whose absolute value is the quotient of the absolute values of the original numbers. 6/-2 = -3 10/-5 = -212/-3 = -4 6/-2 = -3 10/-5 = -212/-3 = -4

19. 4. If one negative real number is divided by another negative real number, the quotient is a positive real number whose absolute value is the quotient of the absolute values of the original numbers. -6/-3 = 2 -8/-2 = 4 -12/-4 = 3 -6/-3 = 2 -8/-2 = 4 -12/-4 = 3

20. We can now consolidate all we have learned about the multiplication and division of signed numbers into two rules:

21. 1. Like Signs: the product or the quotient of two signed numbers that have the same sign is a positive number whose absolute value is the absolute value of the product or the quotient of the absolute values of the original numbers.

22. 2. Unlike Signs: the product or the quotient of two signed numbers that have opposite signs is a negative number whose absolute value is the absolute value of the product or the quotient of the absolute values of the original numbers.

23. More simply: 1. Like signs a positive number. 2. Unlike signs a negative number.

24. Practice: -4(2)-3(-2) -16/2-10/-2 -16/2-10/-2

25. HW: Lesson 9 #1-30 Due Next Time