Lesson 2: Adding Integers

Property

To add the integers $p$ and $q$: begin at zero and draw the line segment (arrow) to $p$.
Starting at the endpoint $p$, draw the line segment representing $q$. Where it ends is the sum $p + q$.
An arrow pointing right is positive, and a negative arrow points left.
Each arrow is a quantity with both length (magnitude) and direction (sign).

Examples

• To calculate $3 + 4$, start at 0, move 3 units right, and then move 4 more units right. You land at 7. So, $3 + 4 = 7$.
• To find $-6 + 4$, start at 0, move 6 units left to $-6$, then move 4 units right. You land at $-2$. So, $-6 + 4 = -2$.
• To compute $-3 + (-5)$, start at 0, move 3 units left to $-3$, then move 5 more units left. You land at $-8$. So, $-3 + (-5) = -8$.

Explanation

Think of adding on a number line as taking a journey. Positive numbers are steps to the right, and negative numbers are steps to the left. Your final position is the sum of the integers.

Property

Same Signs: Add the absolute values and keep the common sign.
Different Signs: Subtract the smaller absolute value from the larger absolute value and use the sign of the integer with the greater absolute value.

Examples

Property

Solve real-world and mathematical problems involving the four operations with rational numbers.
This requires translating a real-world scenario into a mathematical expression using addition, subtraction, multiplication, or division of rational numbers.

Examples

• A baker has $4\frac{1}{2}$ pounds of flour. A cake recipe requires $2\frac{3}{4}$ pounds. How much flour is left? $\frac{9}{2} - \frac{11}{4} = \frac{18}{4} - \frac{11}{4} = \frac{7}{4}$, or $1\frac{3}{4}$ pounds.
• A submarine at the surface dives $20\frac{1}{2}$ meters, then rises $8\frac{1}{4}$ meters. What is its new depth? $-20\frac{1}{2} + 8\frac{1}{4} = -\frac{41}{2} + \frac{33}{4} = -\frac{82}{4} + \frac{33}{4} = -\frac{49}{4}$, or $-12\frac{1}{4}$ meters.
• Three friends share a pizza. Anna eats $\frac{1}{4}$, and Ben eats $\frac{1}{3}$. What fraction of the pizza did they eat combined? $\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$ of the pizza.

Explanation

Rational numbers are used to handle everyday tasks like measuring ingredients, tracking distances, or splitting bills. The first step is to read the problem carefully and decide which of the four basic operations is needed to solve it.