Question

which of these expressions or phrases can be used to represent the model? choose all the correct answers.
$\frac{1}{6}$ of 3
$6 \div 3$
$3 \div \frac{1}{6}$
$\frac{1}{6} \times 3$
number of sixths in 3
number of sixths in 1

Explanation:

Step1: Analyze " $\frac{1}{6}$ of 3"

"Of" in math means multiplication, so $\frac{1}{6}$ of 3 is $\frac{1}{6} \times 3$, which is a valid representation.

Step2: Analyze " $6 \div 3$"

This is 2, which doesn't relate to the model (about sixths and 3), so it's invalid.

Step3: Analyze " $3 \div \frac{1}{6}$"

Dividing by a fraction is multiplying by its reciprocal, so $3 \div \frac{1}{6}=3\times6 = 18$, but the number of sixths in 3: since 1 has 6 sixths, 3 has $3\times6=18$ sixths, and $3\div\frac{1}{6}=3\times6 = 18$, so this is valid.

Step4: Analyze " $\frac{1}{6} \times 3$"

As in step 1, this is the same as $\frac{1}{6}$ of 3, valid.

Step5: Analyze "number of sixths in 3"

Since 1 whole has 6 sixths, 3 wholes have $3\times6 = 18$ sixths. And $3\div\frac{1}{6}=18$, so this phrase is valid.

Step6: Analyze "number of sixths in 1"

Number of sixths in 1 is 6, which is different from the model (about 3), so invalid.

Answer:

$\frac{1}{6}$ of 3, $3 \div \frac{1}{6}$, $\frac{1}{6} \times 3$, number of sixths in 3 (corresponding to the options: the first, third, fourth, and the fifth option (number of sixths in 3))