standard_description: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
problem_activity: Narrative
The purpose of this activity is for students to transition from reasoning about division concretely or visually (using base-ten diagrams) to doing so more abstractly (by writing equations). It also reinforces the connections between multiplication and division.
Students make sense of three different strategies of dividing 78 by 3 and attend to the connections between the visual and numerical representations of the same quotient. As they do so, they practice reasoning quantitatively and abstractly (MP2).
During the synthesis, discuss how place value units play a role in all three strategies.
MLR8 Discussion Supports.
Synthesis: Display Lin, Priya, and Tyler’s strategies. As students share their observations, annotate the display to illustrate connections. For example, annotate where students see the divisor, dividend, and quotient on each diagram.
Advances: Listening, Representing
Launch
Groups of 2
“Take a couple minutes to make sense of Lin, Priya, and Tyler’s work.”
2 minutes: quiet think time
Activity
“Work with your partner to make sense of Lin, Priya, and Tyler’s work and to complete the activity.”
7–8 minutes: partner work time
Monitor for students who:
make connections between the equations and the base-ten diagram
recognize how Tyler’s equations and reasoning are different from Priya’s equations
Identify students who can explain these connections or distinctions to share during the synthesis.
Student Facing
Lin, Priya, and Tyler found the value of
\(78 \div 3\)
. Their work is shown. Make sense of each student’s work.
Lin
[IMG]
Priya
\(\begin{align} 3\times 10&= 30\\ 3\times 10&= 30\\ 3\times \phantom{0}6 &= 18 \\ \overline {\hspace{5mm}3 \times 26} &\overline {\hspace{1mm}=78 \phantom{000}} \end{align}\)
Tyler
\(\begin{align} 3\times 20&= 60\\ 3\times \phantom{0}6 &= 18 \\ \\20 + 6 &= 26 \end{align}\)
How are the three students’ work alike?
How are they different?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite students to share how Lin, Priya, and Tyler’s work are alike and how they are different.
Select previously identified students to share additional connections that they noticed.
“Why does it make sense that Priya and Tyler wrote multiplication equations to find the value of a quotient?” (Multiplication and division can represent the same equal-groups situation. To divide 78 by 3 is to find how many are in each of 3 groups or how many groups of 3 there are. We can multiply up to 78 to find the answer.)
Consider asking: “What new ideas about dividing numbers did you learn and would like to try? Talk to a new partner about why you’d like to try them.”


aligns: no