How do you multiply integers on a number line?

Multiplying integers on a number line involves visualizing repeated addition or subtraction based on the signs of the integers.

Understanding the Basics

• Positive Integer Multiplication: This can be thought of as repeated addition. For example, 3 x 2 means adding 2 to itself 3 times, starting from 0 and moving to the right.
• Negative Integer Multiplication: This involves understanding the concept of "opposite direction."

Steps for Multiplying Integers on a Number Line

1. Identify the Factors: Determine the two integers you are multiplying.
2. Start at Zero: Always begin at zero on the number line.
3. Determine the Direction and Magnitude:
   • The first number indicates how many "jumps" you will make.
   • The second number indicates the size and direction of each jump. If the second number is positive, you jump to the right. If it's negative, you jump to the left.
4. Positive x Positive: Move to the right, adding the second number to itself the number of times specified by the first number. Example: 3 x 4 means jump 4 units to the right, 3 times. You end up at +12.
5. Positive x Negative: Move to the left (because you are adding a negative number repeatedly) the number of times specified by the positive integer. Example: 3 x (-4) means jump 4 units to the left, 3 times. You end up at -12.
6. Negative x Positive: This is conceptually the same as Positive x Negative, due to the commutative property of multiplication (a x b = b x a). So, (-3) x 4 is the same as 4 x (-3). Thus, start at zero and jump 3 units to the left 4 times.
7. Negative x Negative: This is the trickiest. Think of the first negative number as indicating the opposite of what you would normally do. So, (-3) x (-4) means the opposite of adding -4 three times. Adding -4 three times would be moving to -12. The opposite of that is +12. You can visualize this as removing groups of -4. You jump 4 units to the right, 3 times. You end up at +12.

Examples

• 3 x 2: Start at 0, jump 2 units to the right, 3 times. End at +6.
• 3 x (-2): Start at 0, jump 2 units to the left, 3 times. End at -6.
• (-3) x 2: Start at 0, jump 2 units to the left, 3 times. End at -6.
• (-3) x (-2): Start at 0. You're removing sets of -2. So jump 2 units to the right, 3 times. End at +6.

Summary

Multiplying integers on a number line helps visualize the process as repeated addition (or subtraction) and the concept of direction. Remember that multiplying by a negative number can be interpreted as taking the opposite direction on the number line.