In this activity, students use base-ten diagrams to divide two whole numbers that result in terminating decimal quotients. In the given problems, the ones cannot be placed into equal groups of whole numbers without a remainder and must be decomposed into tenths. Students see that this process is conceptually no different than decomposing hundreds into tens or tens into ones so that all the pieces can form equal-size groups.

The goal of the discussion is to highlight that the same reasoning process is involved when we divide two numbers, regardless of whether the quotient is or is not a whole number.

Consider displaying Elena’s diagram for and Mai’s diagram for .

Ask questions such as:

• “What’s the same about Mai’s reasoning for finding and Elena’s process for finding  ?” (They both represented the pieces by place value and put them into equal-size groups. They both had to decompose one or more pieces into 10 smaller units.)
• “What’s different?” (Elena decomposed a ten into 10 ones. Mai decomposed a one into 10 tenths. In Mai’s division, the quotient is not a whole number.)

The big new idea here is that sometimes division of whole numbers does not end when we get to the ones place. When the whole-number amount can no longer be distributed into equal groups, we can first decompose them into tenths, hundredths, or other smaller base-ten units and then distribute them.

In this activity, students analyze several diagrams that represent a decimal being divided by a whole number. They interpret and explain the presence or arrangement of base-ten units in several stages of reasoning, including in a final diagram, which shows the quotient. As students decompose and distribute base-ten pieces into equal-size groups and think about the meaning of each piece, students practice reasoning abstractly and quantitatively (MP2).

Some students may stop dividing when they reach a remainder rather than decomposing the remainder into smaller units. Remind them that they can continue to divide the remainder by decomposing and to refer to Elena’s worked-out example or those from earlier lessons, if needed.

Invite a student or a group to share their explanation of Elena’s diagrams. If possible, display and mark up the second given diagram to reflect students’ explanation and to arrive at the same result as shown in the final diagram.

Emphasize that anytime Elena had base-ten units that can’t be distributed equally into 4 groups, she decomposed each into 10 of the next smaller unit and distributed those. She kept going until there were no pieces remaining.

If time permits, use Critique, Correct, Clarify to give students an opportunity to improve a sample written prediction about the value of , by correcting errors, clarifying meaning, and adding details.

• Display this first draft:

  “The value of would be a decimal with a whole number and 2 tenths because the 8 tenths in 53.8 are divided into 4 groups, which means 2 tenths in each group.”
• Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft.
• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.