Illustrate the quotient by using the given concept of division. Sketch any new pieces that are necessary to show regrouping.



 `396\div132`, using the measurement (subtractive) concept

The measurement (subtractive) concept of division involves repeatedly subtracting the divisor from the dividend until we reach zero or a remainder smaller than the divisor. To illustrate this for \( 396 \div 132 \), we will repeatedly subtract \( 132 \) from \( 396 \) and count how many times we can subtract it fully.

### Step-by-Step Process:

1. **Initial Division**: Start with \( 396 \). Subtract \( 132 \).
   - \( 396 - 132 = 264 \)
   
2. **Second Subtraction**: Take the result \( 264 \). Subtract \( 132 \).
   - \( 264 - 132 = 132 \)
   
3. **Third Subtraction**: Take the result \( 132 \). Subtract \( 132 \).
   - \( 132 - 132 = 0 \)
   
Since we reached 0, we stopped subtracting. We can count how many times we subtracted \( 132 \). The answer is 3 times.

Thus, \( 396 \div 132 = 3 \).

### Sketch of Regrouping:

In the sketch, you would visually represent this subtraction process by:

1. Drawing a bar of length representing \( 396 \).
2. Subdividing this bar into equal segments, each representing the value \( 132 \).
3. Label each subtraction step, showing how the bar diminishes in size until there is no remainder.

To summarize:
- You start with a full bar representing 396.
- The bar is split into three equal parts, each labeled \( 132 \).
- Each division step shows that after three equal divisions of \( 132 \), the full length of \( 396 \) is exhausted.

Would you like further clarification or a more detailed visual representation? Let me know if you have any questions or want to dive deeper.

### Five Related Questions:
1. How does the measurement concept differ from the partitive concept of division?
2. Can you use the measurement method to divide non-whole numbers, like \( 9.6 \div 1.2 \)?
3. How does repeated subtraction work when there is a remainder, such as \( 400 \div 132 \)?
4. Can you apply this method to larger numbers like \( 792 \div 132 \)?
5. What is the connection between the measurement division and long division?

### Tip:
When using the measurement concept, always check if the divisor fully fits into the dividend before starting the subtraction process—this helps avoid errors in cases of remainders.

Explore how to illustrate the division problem 396 ÷ 132 using the measurement (subtractive) concept. Step-by-step explanation and sketching techniques included. Suitable for Grades 3-5 mathematics education.

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