Problem 20 - Algebra 1

Question

Use a number line to find the sum. $$-5+8+\left(-3 \frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The sum of -5, 8, and \(-3 \frac{1}{2}\) is -0.5

Step 1: Plot the first value on number line

Start by plotting the first value which is -5 on the number line.

Step 2: Adding positive value

Next, add 8 by moving to the right of the number line from -5. This will land at +3.

Step 3: Adding negative value

Then, add \(-3 \frac{1}{2}\) by moving to the left, corresponding to subtracting \(3 \frac{1}{2}\). The final position on the number line will be at -0.5.

Key Concepts

Understanding the Number Line

The number line is a visual representation of numbers on a straight line, where each point on the line corresponds to a number. It's a fundamental tool for understanding basic mathematical concepts, particularly operations like addition and subtraction.

To use a number line for addition, you start by plotting the first number at its corresponding location on the line. If the number is positive, it will be to the right of zero, and if it's negative, to the left. When adding or subtracting, you 'move' along the line. Adding a positive number means moving to the right, and adding a negative number means moving to the left, equivalent to subtracting the absolute value of that number. It is helpful to think of the number line as a horizontal road, where moving involves either stepping forwards or backwards, depending on the sign and magnitude of the number you're working with.

Adding Integers

When adding integers, the process can be visualized using the number line. Integers include all whole numbers, both positive and negative, as well as zero. To add integers, you find the first number on the number line and then 'walk' in the direction indicated by the second number.

For example, if you're adding a positive integer, you would move to the right. Conversely, adding a negative integer means moving to the left. The number of steps you move correlates to the absolute value of the integer you're adding. For adding multiple integers, continue this process consecutively, starting from where the last addition left you. It's important to keep track of your starting point and each movement, as this ensures accuracy in finding the sum.

Working with Negative Numbers

Negative numbers are those less than zero, and they are usually represented to the left of zero on a number line. They play a key role in various mathematical operations, including adding integers. When you're adding negative numbers, you're essentially subtracting their absolute value from the starting number.

Negative numbers can seem counterintuitive since they 'reverse' the direction of addition. Remember, moving to the left on the number line decreases the value, reflecting subtraction, even when the operation is framed as addition. It's useful to think of negative numbers like debt; if you 'add' debt, you're really losing money. Therefore, adding negative numbers is mathematically equivalent to losing or subtracting the corresponding positive value. Understanding negative numbers and their behavior during addition helps prevent errors and simplifies more complex calculations.