• Greater than 1. Sample reasoning:
  • There are two $\frac{1}{4}$s in $\frac{1}{2}$.
  • $2 \boldcdot \frac{1}{4}=\frac{1}{2}$
• Greater than 1. Sample reasoning:
  • The division can mean “How many $\frac{3}{4}$s are in 1?” and there is a little more than 1 group of $\frac{3}{4}$ in 1.
  • $\frac{3}{4} \boldcdot \frac{4}{3} = 1$, and $\frac{4}{3}$ is greater than 1.
• Less than 1. Sample reasoning:
  • The divisor $\frac{7}{8}$ is greater than the dividend $\frac{2}{3}$, so the quotient is not quite 1 whole.
  • There is less than 1 group of $\frac{7}{8}$ in $\frac{2}{3}$.
• Greater than 1. Sample reasoning:
  • $\frac{7}{8}$ is closer to 1 than $\frac{3}{5}$ is, so $2\frac{7}{8}$ is greater than $2\frac{3}{5}$. Since the dividend is greater than the divisor, the quotient is greater than 1.
  • $\frac{7}{8}$ is greater than $\frac{3}{4}$ while $\frac{3}{5}$ is less than $\frac{3}{4}$, which means that $\frac{7}{8}$ is greater than $\frac{3}{5}$. It also means that $2\frac{7}{8}$ is greater than $2\frac{3}{5}$ and there is more than one group of $2\frac{3}{5}$ in $2\frac{7}{8}$.

To involve more students in the conversation, consider asking:

• “Who can restate $\underline{\hspace{.5in}}$’s reasoning in a different way?”
• “Did anyone use the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $\underline{\hspace{.5in}}$’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

Highlight that we can estimate the reasonableness of our answers to division questions by thinking about how the dividend compares to the divisor. We can also use the relationship between multiplication and division to check our answers. For instance,  $\frac12 \div \frac14 = ?$ corresponds to $? \boldcdot \frac14 =\frac{1}{2}$, so we can multiply the quotient and $\frac{1}{4}$ to see if it gives the product of $\frac{1}{2}$. 

If the idea of estimating quotients by using benchmark fractions does not come up, discuss it with students. For instance, if students are unsure how $\frac{7}{8}$ and $\frac{3}{5}$ compare, prompt them to think of a familiar fraction that is close to both numbers and to compare each number to that benchmark instead. Ask students: “How does $\frac{7}{8}$ compare to $\frac{3}{4}$?” and “How does $\frac{3}{5}$ compare to $\frac{3}{4}$?”