Write a “how-many-units-in-1-group” word problem for the equation 4÷ 3 2 ​ =?. Use your problem along with a math drawing, table, or double number line to explain why it makes sense to solve 4÷ 3 2 ​ by “inverting and multiplying”—in other words, by multiplying 4 by 2 3 ​ .

Word Problem:

Sarah has 4 meters of ribbon, and she wants to cut it into pieces that are each \( \frac{3}{2} \) meters long. How many pieces of ribbon can she cut?

Explanation:

To solve this problem, we need to determine how many \( \frac{3}{2} \) meter pieces can fit into 4 meters. This can be represented by the equation:

To divide by a fraction, we can use the method of “inverting and multiplying.” This means we multiply by the reciprocal of \( \frac{3}{2} \), which is \( \frac{2}{3} \). Thus, we rewrite the equation as:

Steps:

1. Identify the total length of ribbon: 4 meters.
2. Identify the length of each piece: \( \frac{3}{2} \) meters.
3. Set up the division: \( 4 \div \frac{3}{2} \).
4. Invert the fraction and multiply: \( 4 \times \frac{2}{3} \).

Math Drawing:

Imagine a number line where 0 to 4 meters is marked. Each \( \frac{3}{2} \) meter piece can be represented as a segment on this line:

• 0 to \( \frac{3}{2} \) (1st piece)
• \( \frac{3}{2} \) to 3 (2nd piece)
• 3 to 4 (remaining part)

Table Representation:

| Pieces of Ribbon | Length of Each Piece (m) | Total Length Used (m) |
|——————|————————–|————————|
| 1 | \( \frac{3}{2} \) | \( \frac{3}{2} \) |
| 2 | \( \frac{3}{2} \) | 3 |
| 3 | \( \frac{3}{2} \) | Not enough (4 meters) |

Conclusion:

By multiplying \( 4 \) by \( \frac{2}{3} \), we can easily find how many pieces of \( \frac{3}{2} \) meters fit into 4 meters. The calculation gives us:

This means Sarah can cut 2 full pieces of ribbon, with some leftover. Thus, using the method of inverting and multiplying makes sense in this context.