Question

Marissa claims that stacking 6 blocks with dimensions of One-half times 1 times 1 will give the same volume as stacking 9 blocks with dimensions of One-third times 1 times 1. Which statements about her claim are true? Check all that apply.
Six of the one-half cubes would have a volume of 3 cubic units.
Nine of the one-third cubes would have a volume of 6 cubic units.
Since One-half and One-third do not have common factors, stacks of one-half cubes and stacks of third-cubes could never have the same volume.
Three of the one-third cubes will make 1 unit cube.
Two of the one-half cubes will make 1 unit cube.
Both stacks will have a volume of 3 cubic units.

Answer 1

Let's analyze Marissa's claim step-by-step by calculating the volumes of the blocks and examining the validity of each statement.

1. **Volume of one block**:
- For the block with dimensions \( \frac{1}{2} \times 1 \times 1 \):
\[
\text{Volume} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \text{ cubic units}
\]
- For the block with dimensions \( \frac{1}{3} \times 1 \times 1 \):
\[
\text{Volume} = \frac{1}{3} \times 1 \times 1 = \frac{1}{3} \text{ cubic units}
\]

2. **Volume of the stacks**:
- For 6 blocks of \( \frac{1}{2} \) cubes:
\[
\text{Total Volume} = 6 \times \frac{1}{2} = 3 \text{ cubic units}
\]
- For 9 blocks of \( \frac{1}{3} \) cubes:
\[
\text{Total Volume} = 9 \times \frac{1}{3} = 3 \text{ cubic units}
\]

3. **Evaluating the statements**:

- **"Six of the one-half cubes would have a volume of 3 cubic units."**
- True. We calculated this volumetric total as \( 3 \) cubic units.

- **"Nine of the one-third cubes would have a volume of 6 cubic units."**
- False. We calculated this volumetric total as \( 3 \) cubic units, not \( 6 \).

- **"Since One-half and One-third do not have common factors, stacks of one-half cubes and stacks of third-cubes could never have the same volume."**
- False. Even though \( \frac{1}{2} \) and \( \frac{1}{3} \) do not have common factors, they can still produce the same total volume, as evidenced by their individual volumes at the stack totals we calculated.

- **"Three of the one-third cubes will make 1 unit cube."**
- True.
\[
3 \times \frac{1}{3} = 1 \text{ unit}
\]

- **"Two of the one-half cubes will make 1 unit cube."**
- True.
\[
2 \times \frac{1}{2} = 1 \text{ unit}
\]

- **"Both stacks will have a volume of 3 cubic units."**
- True. We calculated that both stacks indeed have a total volume of \( 3 \) cubic units.

**Final true statements** are:
- Six of the one-half cubes would have a volume of 3 cubic units.
- Three of the one-third cubes will make 1 unit cube.
- Two of the one-half cubes will make 1 unit cube.
- Both stacks will have a volume of 3 cubic units.