Problem 36 - Prealgebra

Question

Complete the following tables. $$\begin{array}{ccc} \hline \begin{array}{c} \text { First } \\ \text { Number } \\ a \end{array} & \begin{array}{c} \text { Second } \\ \text { Number } \\ b \end{array} & \begin{array}{c} \text { Their } \\ \text { Sum } \\ a+b \end{array} \\ \hline-5 & 3 \\ -5 & 4 \\ -5 & 5 \\ -5 & 6 \\ -5 & 7 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Fill the sum column: -2, -1, 0, 1, and 2.

Step 1: Understand the Given Information

We have two columns labeled 'First Number' and 'Second Number', and third column labeled 'Their Sum'. Our goal is to find the sum of two numbers for each row where the 'First Number' is always -5. Each row has a different 'Second Number' value, ranging from 3 to 7.

Step 2: Calculate the Sum for the First Row

For the first row, the 'First Number' is -5 and the 'Second Number' is 3. The sum is calculated as follows: \[-5 + 3 = -2\].

Step 3: Calculate the Sum for the Second Row

For the second row, the 'First Number' is -5 and the 'Second Number' is 4. The sum is calculated as follows: \[-5 + 4 = -1\].

Step 4: Calculate the Sum for the Third Row

For the third row, the 'First Number' is -5 and the 'Second Number' is 5. The sum is calculated as follows: \[-5 + 5 = 0\].

Step 5: Calculate the Sum for the Fourth Row

For the fourth row, the 'First Number' is -5 and the 'Second Number' is 6. The sum is calculated as follows: \[-5 + 6 = 1\].

Step 6: Calculate the Sum for the Fifth Row

For the fifth row, the 'First Number' is -5 and the 'Second Number' is 7. The sum is calculated as follows: \[-5 + 7 = 2\].

Step 7: Complete the Table

Now we can fill in the third column with our calculated sums for each row: - Row 1: -2 - Row 2: -1 - Row 3: 0 - Row 4: 1 - Row 5: 2

Key Concepts

Addition

Addition is one of the four fundamental arithmetic operations. It's a simple process of finding the total or combined amount of two numbers or more. When you add numbers, you are essentially bringing them together, which leads to a larger number, unless you are working with negative numbers.

• To add two numbers, line them up according to their place values, such as units, tens, hundreds, etc.
• Add each column of numbers starting from the rightmost digit and move leftward.
• Carry over any value more than ten to the next higher column.

Addition is written with the plus sign (+), connecting the numbers that are being added to produce a single total or sum. This makes it a very intuitive and essential operation for everyday calculations and higher-level math.

Integers

Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. Integers are straightforward and allow you to perform a variety of operations, like addition, without dealing with complicated parts or halves.
Examples of integers include:

• Positive integers like 1, 2, 3, etc.
• Negative integers such as -1, -2, -3, etc.
• Zero, which is considered an integer.

Working with integers can simplify many mathematical operations. Especially when using negative integers, addition becomes a matter of counting backwards. This understanding is particularly useful when calculating sums that involve negative numbers, as shown in the example exercise.

Number Line

A number line is a visual representation of numbers on a straight line. It can help you understand the positions of numbers, especially when dealing with positive and negative integers.
To use a number line effectively:

• Start at zero in the center, with positive numbers growing to the right and negative numbers to the left.
• To add a positive number, move to the right; to add a negative number, move to the left.
• For example, to calculate \(-5 + 3\), start at -5 on the number line and move three steps to the right, landing on -2.

Using a number line can make it easier to visualize arithmetic operations, enabling a clear connection between numbers and their sums, as shown in the exercise.

Sum Calculation

Calculating the sum means finding the total amount resulting from the addition of two or more numbers. This is what you practiced in the exercise by computing the sums for each pair of integers.
To calculate the sum effectively:

• Identify each number that needs to be added. Ensure their signs (positive or negative) are clear.
• Apply the basic rule: a positive and a negative number undergo subtraction, taking the sign of the larger absolute number.
• Use the simplified process: when both numbers are negative, simply add the absolute values and keep the negative sign. For example, \(-5 + 3\) becomes \(3\) steps rightward from \(-5\), equaling \(-2\).

These concepts will be critical in dealing with more complex operations and are grounded in practice by exercises such as this one.