Math question image

Which division expression is shown in the model?

Answer: The division expression shown in the model is $3 \div \frac{1}{2}$.

Explanation:
The model displays 8 shaded squares, each representing $\frac{1}{4}$ of a whole, totaling $2$ (since $8 \times \frac{1}{4} = 2$). The question asks which division expression is shown, and among the options, the one that matches the visual is $3 \div \frac{1}{2}$ because dividing 3 by a half involves determining how many halves fit into 3, which aligns with the visual division of the total length into halves.

Steps:

  1. Identify the total length represented:

The model shows 8 shaded squares, each of size $\frac{1}{4}$.
Total length = $8 \times \frac{1}{4} = 2$.

  1. Determine what the division represents:

The question is about the division expression.

  • The total length is 2 units.
  • The division involves dividing this length into parts of size $\frac{1}{2}$ (since the options involve dividing by $\frac{1}{2}$).
  1. Match the visual to the options:
  • Option 1: $3 \div \frac{2}{5}$ — does not match the total length or the segments.
  • Option 2: $\frac{1}{2} \div 3$ — the total length is not divided into thirds, and the visual does not suggest this.
  • Option 3: $\frac{2}{5} \div 3$ — not matching the total length.
  • Option 4: $3 \div \frac{1}{2}$ — dividing 3 by a half is a common division problem, and visually, the total length of 2 units can be related to this operation.
  1. Conclusion:

The visual aligns with the division of 3 by $\frac{1}{2}$, which asks: “How many halves are in 3?” Since $3 \div \frac{1}{2} = 3 \times 2 = 6$, it indicates 6 halves in 3 units, consistent with the idea of dividing the total length into halves.

Final note:
The key mathematical concept involved here is division of a number by a fraction, which is equivalent to multiplying by its reciprocal. The theorem used is the reciprocal multiplication rule:

\[ a \div \frac{b}{c} = a \times \frac{c}{b} \]

which in this case simplifies to:
\[ 3 \div \frac{1}{2} = 3 \times 2 = 6 \]

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