Integers: Multiplication & Division

I. Algorithm for Multiplying or Dividing Integers

Use the standard algorithm for multiplying whole numbers
1) When multiplying or dividing  integers that have the same sign the result is always positive. Simply multiply or divide the absolute value of each number and make the answer positive.

-Positive x Positive = Positive

-Negative x Negative = Positive

2) When multiplying or dividing integers that have different signs the result is always negative. Multiply or divide the absolute value of each number and make the answer negative.

-Positive x Negative = Negative

-Negative x Positive = Negative

II. Conceptual Models
1. Number Line
We can use the number line to help conceptually understand the rules for multiplying and dividing integers.

positive x positive = positive
Multiplying positive integers is no different than multiplying whole numbers. For example, 3 x 4 can be thought of as moving 4 units to the right 3 times.

Since division is the inverse operation of multiplication, we can reverse the above problem. Starting at 12, we move in groups of 4 to the left until we reach 0. We then count the number of groups or movements to reach the answer of 3.

positive x negative = negative
Multiplying 3 x (-4) can be thought of as "3 groups of -4." Starting at 0 on the number line, we move 4 units to the left 3 times, or as your teacher might say, you take 3 jumps of -4 to arrive at -12.

negative x positive = negative
With (-3) x 4 the 4 is our value (multiplicand) and the -3 is our action (multiplier). The negative sign in the multiplier essentially means do the opposite (go in the opposite direction). So we are“−3 groups of 4” means the opposite movement: start at 0, move 4 units to the left 3 times to arrive at -12..

 negative x negative = positive

As a negative number is the opposite of a positive number, the opposite of a negative number is positive. Hence, a negative times a negative equal a positive. For instance, to solve (-3) x (-4) we can start by using the number line above. As with the problem 3 x (-4) we can view this problem as repeated addition where we are going to move 4 units to the left 3 times leaving us at -12. But we haven't accounted for the negative sign in front of the 3 yet. To do so, we take the opposite of -12 (a negative of a negative is a positive) which leaves us at positive 12 for our answer.


2. Using Color Counters

This video does a good job of explaining ways of conceptualizing integer multiplication and division.

Mathispower4u Video

3. Division Based on Divisor-Dividend Size

In the whole number division section I included some rules on how the quotient of a division problem would result based on the size of the divisor compared to the dividend. Those rules where:

1) Divisor smaller than dividend - The quotient is greater than 1.

2) Divisor equal to dividend - The quotient is 1

3) Divisor larger than dividend - The quotient is between 0 and 1.

These rules work for integers if you change all the values to absolute values. So the revised rules would be:

1) Absolute value of Divisor is smaller than absolute value of the dividend - The absolute value of the quotient is greater than 1.

(e.g. -12 ÷ 3 becomes |-12| ÷ |3| so |quotient| is greater than 1

2) Absolute value of the Divisor is equal to the absolute value of the dividend - The absolute value of the quotient is 1

3) Absolute value of the Divisor larger than absolute value of the dividend - The absolute value of the quotient is between 0 and 1.

(e.g. -3  ÷ 12 becomes |-3| ÷ |12| so |quotient| is between 0 and 1

III. Properties



Practice:
IXL: Multiplying and Dividing Integers

MathGames: Multiply and Divide Integers

Reference
The Monterey Institute: Multiplying and Dividing Real Numbers
The University of Georgia: Why is a negative number times a negative number a positive number
Math is Fun: Multiplying Negatives
The Math Forum@Drexel: Negative x Negative = Positive
https://kobrienmath.weebly.com/lesson-blogs/conceptually-understanding-operations-with-integers

https://www.maneuveringthemiddle.com/teaching-multiplying-and-dividing-integers/