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

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

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

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

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.

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

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

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.

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.

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

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

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

Inverse Operations*: When one operation undoes another operation.

7 + 3 – 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.

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

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 = /5 = -2-12/3 = -4 -6/2 = /5 = -2-12/3 = -4

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

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

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

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.

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.

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

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

HW: Lesson 9 #1-30 Due Next Time