## Generalizations About Adding

## Work Time

# Generalizations About Adding

Jack, Marcus, and Lucy made the following generalizations about adding integers.

**Jack's generalization:**

You can use the number line to show $a+b$, where $a$ and $b$ are any two numbers:

- Start at $a$ (which may be positive, negative, or 0).
- From $a$, move a distance of $\left|b\right|$ to the right if $b$ is positive or to the left if $b$ is negative. The number you arrive at on the number line is the answer to $a+b$.

**Marcus's generalization:**

- The sum $a+b$ is negative if both $a$ and $b$ are negative or if the negative number is a greater distance from 0 (has a greater absolute value) than the positive number.
- The sum $a+b$ is positive if both $a$ and $b$ are positive or if the positive number is a greater distance from 0 (has a greater absolute value) than the negative number.

**Lucy's generalization:**

- When you add two numbers with the same sign, add their absolute values and write the sum using the sign of the numbers.
- When you add two numbers with different signs, find the difference between their absolute values and write the difference using the sign of the number that has the larger absolute value.

Who is correct? Explain why.

## Hint:

Look at your equations and number lines from the previous problems.

- Apply each student’s generalization to your work. Does the generalization work?
- Think about how you created each number line. How did the signs of the addends determine what you did on the number line?