2. Building on the Essential Question How is multiplying a decimal by a whole number similar to and different from multiplying two whole numbers?

Multiplying a decimal by a whole number shares several similarities with multiplying two whole numbers, but there are also important differences to consider. Here's a detailed comparison: ### **Similarities** 1. **Basic Multiplication Process:** - **Algorithms Used:** Both operations use standard multiplication algorithms, such as the traditional long multiplication method. - **Properties Applied:** Properties like the **commutative** (e.g., \( a \times b = b \times a \)), **associative** (e.g., \( (a \times b) \times c = a \times (b \times c) \)), and **distributive** (e.g., \( a \times (b + c) = a \times b + a \times c \)) properties are applied in both cases. 2. **Step-by-Step Multiplication:** - **Digit Alignment:** Digits are aligned according to their place values (units, tens, hundreds, etc.) before performing multiplication. - **Partial Products:** Both operations involve calculating partial products and then summing them to get the final result. ### **Differences** 1. **Handling the Decimal Point:** - **Decimal Placement:** When multiplying a decimal by a whole number, after performing the multiplication as if both were whole numbers, you must correctly place the decimal point in the product. The decimal is placed by counting the number of decimal places in the original decimal number. - *Example:* \[ 3.2 \times 4 = 12.8 \] *Calculation Steps:* - Multiply \( 32 \times 4 = 128 \). - Since \( 3.2 \) has one decimal place, place the decimal one place from the right: \( 12.8 \). 2. **Number Representation:** - **Whole Numbers vs. Decimals:** Whole number multiplication involves only integers, whereas multiplying by a decimal introduces fractional parts or tenths, hundredths, etc. - **Precision:** Decimals require attention to precision and may involve rounding, which is not typically a concern with whole numbers. 3. **Conceptual Understanding:** - **Value Interpretation:** Multiplying by a decimal often represents scaling by a fraction of a whole (e.g., 0.5 means half), adding a layer of conceptual understanding compared to whole number multiplication, which deals with complete units. 4. **Potential for More Steps:** - **Adjusting for Decimals:** Ensuring the decimal point is correctly placed adds an extra step to the multiplication process, which isn't needed when both numbers are whole. ### **Example Comparison** - **Multiplying Two Whole Numbers:** \[ 23 \times 5 = 115 \] *Steps:* - Multiply \( 23 \times 5 = 115 \). - **Multiplying a Decimal by a Whole Number:** \[ 4.6 \times 3 = 13.8 \] *Steps:* - Multiply \( 46 \times 3 = 138 \). - Since \( 4.6 \) has one decimal place, place the decimal one place from the right: \( 13.8 \). ### **Key Takeaways** - **Process Similarity:** The fundamental multiplication process remains consistent whether dealing with whole numbers or decimals. - **Attention to Decimals:** The primary difference lies in correctly handling the decimal point to ensure the product accurately reflects the values being multiplied. - **Understanding Context:** Recognizing when and how to apply decimal placement is crucial for accurate calculations involving decimals. By understanding both the shared procedures and the unique considerations when decimals are involved, students can confidently perform and distinguish between these types of multiplication operations.