Problem 35 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The 'Sum' column is: 2, 1, 0, -1, -2.

Step 1: Understanding the Problem

We are given a table with two columns filled: the 'First Number' (\(a\)) is 5, and the 'Second Number' (\(b\)) varies from -3 to -7. We need to find the sum \(a+b\) for each row.

Step 2: Calculating the Sum for Each Row

To complete the 'Sum' column, we add the first number (5) with each corresponding 'Second Number' (\(b\)). Perform this calculation for each pair of \(a\) and \(b\).

Step 3: Calculate Sum for Row 1

For the first row, \(a = 5\) and \(b = -3\). Thus, \(a+b = 5 + (-3) = 2\).

Step 4: Calculate Sum for Row 2

For the second row, \(a = 5\) and \(b = -4\). Therefore, \(a+b = 5 + (-4) = 1\).

Step 5: Calculate Sum for Row 3

For the third row, \(a = 5\) and \(b = -5\). Thus, \(a+b = 5 + (-5) = 0\).

Step 6: Calculate Sum for Row 4

For the fourth row, \(a = 5\) and \(b = -6\). Therefore, \(a+b = 5 + (-6) = -1\).

Step 7: Calculate Sum for Row 5

For the fifth row, \(a = 5\) and \(b = -7\). Thus, \(a+b = 5 + (-7) = -2\).

Step 8: Fill in the Table

Using the calculated sums, fill in the 'Their Sum \(a+b\)' column for each row: 2, 1, 0, -1, and -2 respectively.

Key Concepts

Integer Addition Simplified

Addition with integers is a fundamental aspect of math, especially when it involves both positive and negative numbers. Integers are numbers without any fractional part, and they can be positive, negative, or zero.

• A positive integer added to another positive integer will result in a larger positive number.
• Adding two negative integers results in a larger negative number, as you are moving further down the number line.
• When a positive integer is added to a negative integer, the result depends on which absolute value is larger.

For example, consider adding 5 and -3. Start at +5 on a number line and move left 3 spaces (because of the -3), arriving at 2.
This is how we get the sum of 5 + (-3) = 2. This principle applies to all integer addition involving negatives and positives.

Basic Arithmetic with Integers

Basic arithmetic involves simple operations such as addition, subtraction, multiplication, and division. Here's a closer look at how these operations work with integers, focusing on integer addition.
In integer addition, two cases are primarily considered:

• If the numbers have the same sign (both positive or both negative), you add their absolute values and keep the common sign.
• If the numbers have different signs, subtract the smaller absolute value from the larger and use the sign of the number with the larger absolute value.

An example can clarify: for 5 + (-4), subtract 4 from 5 (considering only absolute values), giving 1. The sign of the result is positive because 5 (positive) has a larger absolute value than -4 (negative).

Prealgebra Calculations Made Easy

Prealgebra is a math course designed to prepare students for algebra. It covers topics such as integers, fractions, and basic operations. Integer addition is a building block in this endeavor.
To solve integer addition problems effectively:

• Identify whether the integers are positive or negative.
• Use a number line visualization to help understand the process.
• Apply rules of arithmetic operations and signs carefully.

Consider the prealgebra operation of 5 + (-7). Visualizing on a number line, start at 5 and move 7 steps left (because of the negative sign), ending at -2. This simple visualization technique can make prealgebra calculations more intuitive and less daunting.