Problem 38 - Prealgebra

Question

Complete the following tables. $$\begin{array}{|c|c|} \hline \text { First } & \text { Second } & \text { Their } \\ \text { Number } & \text { Number } & \text { Sum } \\ x & y & x+y \\ \hline 30 & -20 & \\ \hline-30 & 20 & \\ \hline-30 & -20 & \\ \hline 30 & 20 & \\ \hline-30 & 0 & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The sums are 10, -10, -50, 50, and -30 respectively.

Step 1: Sum of the First Row

The first row in the table has the values 30 and -20 for the first and second numbers respectively. Calculate their sum: \( 30 + (-20) \). The sum is equal to \( 10 \).

Step 2: Sum of the Second Row

The second row in the table has the values -30 and 20. Calculate their sum: \( -30 + 20 \). The sum is equal to \( -10 \).

Step 3: Sum of the Third Row

The third row in the table has the values -30 and -20. Calculate their sum: \( -30 + (-20) \). The sum is equal to \( -50 \).

Step 4: Sum of the Fourth Row

The fourth row in the table has the values 30 and 20. Calculate their sum: \( 30 + 20 \). The sum is equal to \( 50 \).

Step 5: Sum of the Fifth Row

The fifth row in the table has the values -30 and 0. Calculate their sum: \( -30 + 0 \). The sum is equal to \( -30 \).

Key Concepts

Understanding Prealgebra

Prealgebra forms the foundation of algebra and focuses mainly on arithmetic involving basic mathematical operations. It prepares students for more advanced topics by introducing them to expressions and simple equations. In the context of integer addition, prealgebra aims to make students comfortable with adding positive and negative numbers.

By working through exercises like this, you develop skills to manipulate numbers and understand how they interact in different operations. Prealgebra helps in grasping the importance of each number's sign and its impact on calculations. Using examples with integers—like the mix of positive and negative numbers—enables students to see how different values can sum up to a new total. This forms the essential basis for handling complex algebraic equations in the future.

• Identifying numbers' values and signs
• Understanding the relationship between numbers in addition
• Building confidence for advanced algebraic concepts

The Process of Sum Calculation

Sum calculation is a critical aspect of prealgebra. It involves adding two or more numbers to find their total, or sum. In this exercise, the focus is on integer addition, where the task is to add pairs of positive and negative numbers.

To calculate a sum, keep these points in mind: When adding two positive integers, you simply combine their values. If you add a positive with a negative integer, you subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value. When you add two negative integers, you add their absolute values and keep the negative sign.

Let's break this down with examples from the exercise:

• For the pair \( 30 + (-20) \), subtract 20 from 30, resulting in 10, a positive number.
• With \( -30 + 20 \), subtract 20 from 30, since 30 has the larger absolute value, resulting in -10.
• Adding two negatives like \( -30 + (-20) \) yields -50.

Through such practice, sum calculation becomes clear and intuitive.

Exploring Mathematical Operations

Mathematical operations such as addition are cornerstone skills in math and go beyond simple arithmetic to form part of more complex calculations later on. The exercise provided is a beginning step into understanding these operations with integers.

Let's take a closer look at integer addition, which involves combining two numbers that can be positive or negative. Every mathematical operation has rules. For addition:

• When adding two positive numbers, you move to the right on the number line.
• Adding two negatives means moving left, discarding their negative signs before summing them.
• When the numbers have different signs, subtract their absolute values to determine the direction of movement on the number line.

Simple exercises like the given tables allow practice and judgment on how integers affect sums, leading to a stronger understanding of mathematical operations. Mastery of such operations ensures a smooth transition to more advanced topics like algebra.