Problem 63 - Fundamentals of Mathematics

Question

Add and subtract as in dicated. Is the sum of 626 and 1,242 the same as the sum of 1,242 and \(626 ?\) Justify your claim.

Step-by-Step Solution

Answer

Yes, the sums are the same (1,868) due to the commutative property of addition.

Step 1: Understand the Commutative Property

The commutative property of addition states that the order of addends does not change the sum. Therefore, \(a + b = b + a\). We will apply this property to our numbers.

Step 2: Apply the Commutative Property

Calculate \(626 + 1,242\). According to the commutative property, this should equal \(1,242 + 626\).

Step 3: Calculate 626 + 1,242

Add the two numbers in the order provided. \[626 + 1,242 = 1,868\].

Step 4: Calculate 1,242 + 626

Now, add the two numbers in the reverse order to confirm the commutative property. \[1,242 + 626 = 1,868\].

Step 5: Justify Your Claim

Since both sums equal 1,868, this confirms that the sum is the same, regardless of the order. This is evidence of the commutative property of addition.

Key Concepts

Addition

Addition is one of the four fundamental arithmetic operations in mathematics. It combines two or more numbers to form a total sum. The numbers being added together are called addends, and the result is known as the sum. Imagine you have a book club, and after gathering together 10 novels from one member and 15 novels from another, you put them in one stack. All those books added together total 25 novels.

Understanding addition means realizing that it is not just about placing numbers next to each other; it’s about bringing quantities together to see how they sum up. For example, adding 626 and 1,242 gives us 1,868. Here, 626 and 1,242 are the addends, and 1,868 is the sum.

Addition is everywhere; it helps us in splitting bills, calculating distances, counting objects, and even understanding more complex mathematical concepts. It is essential to grasp this basic operation to build on more complicated topics in mathematics.

Problem Solving

Problem solving involving addition means carefully applying mathematical principles to find a solution. In mathematics, discovering patterns or using properties like the commutative property can resolve questions effectively.

When faced with a problem like determining whether 626 + 1,242 is the same as 1,242 + 626, it is important to break down the problem.

• Identify what needs to be calculated.
• Understand the properties of the numbers involved.
• Organize the calculation steps in a logical order.

Applying a strategic approach helps keep the process simple and efficient. Problem solving is more than just arriving at an answer; it's about understanding the method to reach that answer. This ensures we're solving the problem accurately and learning something valuable for similar mathematical challenges you might face in the future.

Mathematical Justification

Mathematical justification is the process of reasoning and explaining why something holds true in mathematics. It validates solutions and ensures that conclusions are based on established principles and properties.

Let's take the commutative property of addition, noted in the exercise, which states that the order of numbers does not change their total. To justify this mathematically, you demonstrate that for two numbers like 626 and 1,242, both orderings result in the same sum.

\[626 + 1,242 = 1,868\]
\[1,242 + 626 = 1,868\]

By confirming the same result from different orders, you provide evidence for the commutative property. Justification not only supports your answer but also reinforces your understanding of mathematical concepts.

It's crucial because it deepens your comprehension of mathematical rules and how to apply them, boosting your confidence in working with numbers and solving problems.