Jackie's Distributive Property Problem A Step-by-Step Solution

When tackling mathematical problems, the distributive property stands out as a fundamental concept, especially when dealing with expressions involving multiplication and addition within parentheses. The distributive property provides a method to simplify such expressions by multiplying a factor outside the parentheses with each term inside. Let's delve deeper into understanding this property and how it applies to the problem at hand. In essence, the distributive property states that for any numbers a, b, and c, the equation a × (b + c) is equivalent to (a × b) + (a × c). This seemingly simple rule forms the bedrock for solving a multitude of algebraic problems and is crucial for simplifying complex expressions. The core idea is to 'distribute' the multiplication across the addition (or subtraction) within the parentheses. To effectively utilize the distributive property, it’s crucial to first identify the factor outside the parentheses and the terms inside. Once identified, each term inside the parentheses is multiplied by the factor outside. This process eliminates the need to perform the addition (or subtraction) within the parentheses first, which is particularly useful when the terms inside are not readily combinable, such as in algebraic expressions with variables. Consider a scenario where we have the expression 4 × (x + 3). Applying the distributive property, we multiply 4 by both x and 3, resulting in (4 × x) + (4 × 3), which simplifies to 4x + 12. This simple example showcases the power of the distributive property in transforming expressions into a more manageable form. The distributive property isn't just a mathematical trick; it's a logical rule that stems from the fundamental principles of arithmetic. It ensures that multiplication is correctly applied across a sum or difference, maintaining the equality of the expression. In more complex scenarios, the distributive property can be used in conjunction with other algebraic techniques to solve equations and simplify expressions. For instance, it might be used to expand expressions before combining like terms or to factor out common factors. Mastery of the distributive property is therefore an essential stepping stone for success in algebra and beyond.

In this specific problem, Jackie is faced with the expression $6 imes (10 + 3)$. This expression is a perfect candidate for applying the distributive property. To effectively use this property, Jackie needs to recognize the components of the expression and how they correspond to the general form a × (b + c). Here, 6 is the factor outside the parentheses (equivalent to a), while 10 and 3 are the terms inside the parentheses (equivalent to b and c, respectively). The distributive property dictates that Jackie should multiply 6 by both 10 and 3 separately and then add the results. This is a crucial step in simplifying the expression and finding the solution. By applying the distributive property, Jackie transforms the original expression into an equivalent form that is easier to compute. Instead of first adding 10 and 3 and then multiplying by 6, she can multiply 6 by 10 and 6 by 3 individually, and then sum the products. This approach not only simplifies the calculation but also demonstrates a deep understanding of mathematical principles. The process of distribution involves breaking down a complex multiplication problem into smaller, more manageable parts. This is particularly useful when dealing with larger numbers or algebraic expressions where mental calculation might be challenging. By distributing the multiplication, Jackie is essentially applying the order of operations in a strategic manner, leveraging the distributive property to her advantage. It’s important to note that the distributive property is not just a shortcut; it's a fundamental property that ensures the mathematical integrity of the expression. It guarantees that the transformed expression is equivalent to the original, meaning they will yield the same result. This equivalence is crucial in maintaining the accuracy of mathematical calculations and problem-solving. Moreover, understanding the distributive property is essential for more advanced mathematical concepts, such as factoring and expanding algebraic expressions. It forms the basis for manipulating equations and solving for unknown variables. Therefore, mastering this property is not just about solving this specific problem; it's about building a solid foundation for future mathematical endeavors.

To determine Jackie's next step, we need to carefully analyze the options provided and see which one correctly applies the distributive property to the expression $6 imes (10 + 3)$. The distributive property, as we've discussed, involves multiplying the factor outside the parentheses (6 in this case) with each term inside the parentheses (10 and 3). This means Jackie needs to multiply 6 by 10 and 6 by 3 separately. Let's examine the given options:

• A. $10 + (6 imes 3)$: This option incorrectly rearranges the terms and only multiplies 6 by 3, neglecting the multiplication with 10. It does not accurately represent the distributive property.
• B. $(6 * 10) * 3$: This option uses multiplication but does not distribute 6 across both 10 and 3. It suggests multiplying 6 by 10 first and then multiplying the result by 3, which is a different operation altogether and violates the distributive property.
• C. $(6 imes 10) + (6 imes 3)$: This option perfectly demonstrates the distributive property. It multiplies 6 by 10 and 6 by 3 separately and then adds the results, which is the correct application of the property.
• D. $(6 + 10)$: This option completely disregards the multiplication and only focuses on addition, which is incorrect in the context of the distributive property.

Therefore, the correct next step for Jackie, based on the distributive property, is to transform the expression into $(6 imes 10) + (6 imes 3)$. This option accurately reflects the distribution of the multiplication across the addition within the parentheses. Choosing this option demonstrates a clear understanding of the distributive property and its application in simplifying mathematical expressions. It's essential for Jackie to select this option to proceed with solving the problem correctly. The distributive property is a powerful tool, and recognizing its proper application is key to success in algebra and beyond.

Option C, $(6 imes 10) + (6 imes 3)$, is the unquestionably correct next step for Jackie. This choice precisely embodies the distributive property, which is the core concept being tested in this problem. By selecting this option, Jackie demonstrates a clear and accurate understanding of how to apply this property to simplify mathematical expressions. The distributive property, as we've thoroughly discussed, allows us to multiply a factor outside parentheses by each term inside the parentheses separately and then combine the results. In this case, the factor outside is 6, and the terms inside are 10 and 3. Option C explicitly shows 6 being multiplied by both 10 and 3, with the products then being added together. This is the hallmark of the distributive property in action. The other options, as we've analyzed, deviate from this principle in various ways. They either neglect the multiplication of 6 with both terms, incorrectly rearrange the terms, or disregard the multiplication altogether. Option C stands out as the only choice that maintains the mathematical integrity of the expression while correctly applying the distributive property. Choosing this option is not just about finding the right answer; it's about understanding the underlying mathematical concept and applying it appropriately. It signifies that Jackie grasps the essence of the distributive property and can use it to transform complex expressions into simpler, more manageable forms. This understanding is crucial for building a strong foundation in algebra and other mathematical disciplines. Moreover, selecting Option C sets Jackie on the right path to solving the problem completely. Once the expression is transformed into $(6 imes 10) + (6 imes 3)$, the next step is simply to perform the multiplications and then the addition, leading to the final solution. This highlights the importance of choosing the correct initial step when applying mathematical properties; it paves the way for a smooth and accurate problem-solving process.

To further solidify understanding, let's reiterate why the other options are incorrect in the context of the distributive property and the expression $6 imes (10 + 3)$. Option A, $10 + (6 imes 3)$, is incorrect because it only multiplies 6 by 3 and neglects the multiplication with 10. The distributive property requires that the factor outside the parentheses be multiplied by each term inside. This option essentially skips a crucial step, leading to an incorrect simplification of the expression. It also misinterprets the order of operations that the distributive property helps to manage. The distributive property is designed to handle multiplication across addition (or subtraction), and this option fails to do so comprehensively. Option B, $(6 * 10) * 3$, is also incorrect because it performs a different operation than what the distributive property dictates. This option suggests multiplying 6 by 10 first and then multiplying the result by 3. While this is a valid mathematical operation in itself, it does not correctly apply the distributive property to the original expression. The distributive property involves distributing the multiplication across the addition within the parentheses, not multiplying sequentially. This option alters the fundamental structure of the expression and leads to a different result. The key distinction is that the distributive property transforms a × (b + c) into (a × b) + (a × c), whereas Option B performs (a × b) × c, which is a different operation altogether. Option D, $(6 + 10)$, is perhaps the most incorrect choice as it completely disregards the multiplication aspect of the problem. This option focuses solely on addition and ignores the factor of 6 outside the parentheses. It demonstrates a fundamental misunderstanding of the distributive property and its purpose. The distributive property is specifically designed to handle situations where multiplication is combined with addition (or subtraction), and this option overlooks the multiplication entirely. Choosing this option would indicate a lack of understanding of the basic principles of order of operations and how they apply to expressions involving parentheses and multiplication. In summary, each of the incorrect options fails to accurately apply the distributive property in its intended manner. They either miss crucial steps, perform different operations, or disregard the multiplication altogether. This detailed analysis underscores the importance of understanding the distributive property thoroughly and applying it correctly to simplify mathematical expressions.

In conclusion, the problem presented to Jackie serves as an excellent illustration of the distributive property and its application in simplifying mathematical expressions. The correct next step, as we've established, is C. $(6 imes 10) + (6 imes 3)$. This option accurately reflects the distribution of the multiplication across the addition within the parentheses, which is the essence of the distributive property. By understanding and applying this property, Jackie can transform the original expression into a more manageable form, paving the way for a straightforward solution. The distributive property is not just a mathematical trick; it's a fundamental principle that stems from the core rules of arithmetic. It ensures that multiplication is correctly applied across a sum or difference, maintaining the equality of the expression. This is crucial for accuracy in mathematical calculations and problem-solving. Moreover, mastering the distributive property is essential for more advanced mathematical concepts. It forms the basis for factoring, expanding algebraic expressions, and solving equations. It's a stepping stone to higher-level mathematics and a skill that will be used repeatedly throughout one's mathematical journey. The incorrect options in this problem highlight common misunderstandings of the distributive property. They serve as valuable learning opportunities, emphasizing the importance of careful attention to detail and a thorough understanding of mathematical principles. By analyzing why these options are incorrect, students can solidify their grasp of the distributive property and avoid making similar mistakes in the future. Ultimately, the ability to correctly apply the distributive property is a testament to one's mathematical proficiency. It demonstrates a deep understanding of how numbers and operations interact and the ability to manipulate expressions in a logical and accurate manner. It's a skill that empowers students to tackle complex problems with confidence and precision. Therefore, mastering the distributive property is not just about solving this specific problem; it's about building a strong foundation for future mathematical success. It's about developing a mindset of mathematical reasoning and problem-solving that will serve students well in all their academic endeavors.