Problem 49 - Prealgebra

Question

Write a numerical sentence that illustrates the Commutative Property of Multiplication.

Step-by-Step Solution

Answer

Numerical sentence: \(3 \times 4 = 4 \times 3\), illustrating the commutative property.

Step 1: Understanding the Commutative Property of Multiplication

The commutative property of multiplication states that changing the order of the factors does not change the product. In other words, if you multiply two numbers in different orders, the result will still be the same.

Step 2: Identifying Numbers for the Numerical Sentence

Select two numbers to create the multiplication sentence. For instance, choose the numbers 3 and 4 for this exercise.

Step 3: Writing the First Multiplication Sentence

Create a multiplication sentence using the selected numbers. The first sentence will be: \(3 \times 4\).

Step 4: Writing the Second Commuted Multiplication Sentence

Switch the order of the two numbers to illustrate the commutative property. Write the second sentence as: \(4 \times 3\).

Step 5: Verifying Both Products

Calculate both multiplication sentences. For \(3 \times 4\), the product is 12, and for \(4 \times 3\), the product is also 12, confirming the commutative property holds.

Step 6: Conclusion of the Example

Thus, the numerical sentences \(3 \times 4 = 12\) and \(4 \times 3 = 12\) show that the order of multiplication does not affect the product.

Key Concepts

Understanding Multiplication

Multiplication is a fundamental math operation that represents repeated addition. Imagine you have several groups of the same size, and you want to find out how many items you have in total. For example, if you have 3 groups of 4 apples, you can say that you have a total of 3 times 4 apples.
This is represented as a multiplication sentence:

• The number of groups (3) is called the multiplicand.
• The size of each group (4) is known as the multiplier.
• The result (12) is the product.

So, in numerical terms, it means calculating how many apples you have by adding them: 4 + 4 + 4. Instead of adding repeatedly, you use multiplication for efficiency.

Exploring Mathematical Properties

Mathematical properties are rules or laws that describe relationships between numbers and operations. The commutative property, specific to addition and multiplication, states that the order of numbers does not affect the result.
For multiplication, if you have two numbers, say A and B, then according to the commutative property:

• \(A \times B = B \times A\)
• This means that swapping the numbers around doesn't change the outcome.

This property highlights foundational consistency in mathematics, ensuring that equations can be simplified or rearranged without altering their meaning or results.

Crafting Numerical Sentences

Numerical sentences use numbers and operations to represent mathematical ideas clearly and concisely. An excellent way to illustrate a property like commutative multiplication is through these sentences.
Let's see an example: You choose two numbers, 3 and 4, and write two multiplication sentences:

• First, write: \(3 \times 4 = 12\)
• Then, switch the order: \(4 \times 3 = 12\)

Both give the same product, showing that the order in multiplication doesn’t change the result. These sentences not only prove the commutative property but also help in verifying understanding by yielding consistent verdicts across arrangements.