Mark used a number line to model dividing 2

Question

3
by 3. He divided the number line from 0 to 1 in 3 equal sections, and then divided each of those sections into 2 equal parts. Mark says 2
3
÷ 3 is 1
3
.

Is his answer correct? Explain why or why not.

Answer 1

Mark's reasoning for dividing \( \frac{2}{3} \) by \( 3 \) using a number line needs clarification. Let’s analyze his approach step by step:

1. Understanding \( \frac{2}{3} \): The fraction \( \frac{2}{3} \) represents 2 parts out of a total of 3 equal parts of 1. So, on a number line, \( \frac{2}{3} \) lies two-thirds of the way between 0 and 1.

2. Dividing by 3: When you divide \( \frac{2}{3} \) by \( 3 \), you are essentially asking, "What is \( \frac{2}{3} \) split into 3 equal parts?" To find \( \frac{2}{3} \div 3 \), you can rewrite it as \( \frac{2}{3} \times \frac{1}{3} \).

3. Calculating: Performing the multiplication: \[ \frac{2}{3} \times \frac{1}{3} = \frac{2 \times 1}{3 \times 3} = \frac{2}{9}. \]

4. Mark's Conclusion: Mark concluded that \( \frac{2}{3} \div 3 = \frac{1}{3} \). However, this is incorrect as we calculated that it should be \( \frac{2}{9} \).

Conclusion:

Mark's answer of \( \frac{1}{3} \) is incorrect. The correct result of dividing \( \frac{2}{3} \) by \( 3 \) is \( \frac{2}{9} \). His method of dividing the number line was not appropriate for obtaining the correct answer because he did not account for how dividing \( \frac{2}{3} \) into 3 equal parts would affect the end result. Instead, it should have been divided into smaller segments, leading to finding \( \frac{2}{3} \div 3 \) explicitly as \( \frac{2}{9} \).