In this activity, students use long division to find the decimal expansion for three different fractions. They see that two of the three fractions result in repeating decimals. As students describe and compare the quotients and , they attend to precision of language (MP6).

The purpose of this discussion is to help students make sense of the value of repeating decimals. Direct students’ attention to the reference created using Collect and Display. Ask students to share how they determined which fraction has the greatest value. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases, such as “repeating decimal,” “terminating decimal,” “place value,” “hundredths place,” “thousandths place,” “bar over ___.”

The key takeaway is that 0.36, , and are different values, even though their decimal representations look quite similar.

In this activity students match tape diagrams, verbal descriptions, and equations. The descriptions express the increase or decrease as a fraction of the original amount, while the equations express the scale factor as a decimal. This gives students more practice converting fractions to decimals, including repeating decimals. Then students create their own diagram to represent one of the equations that didn’t have a match.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3). 

The purpose of this discussion is to emphasize the connection between the numbers in the description and the numbers in the equation. First, ask students to share which description and equation they matched with each diagram. Encourage students to agree or disagree and to restate other students’ reasoning in their own words.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to Diagram A by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:

  “Diagram A shows an increase by , so I matched it with .”

  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. 
• Here is an example of a second draft:

  “In Diagram A, if is the original, then is an increase by , but the equation wasn’t on the list. If we view as the original, then is a decrease by , which matches with .

• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

If time permits, invite students to share the diagram they created. Ask other students to identify the equation that represents the same relationship.

In this partner activity, students take turns matching situations and equations. The situations are the same as students saw in a previous activity, but this time the equations use decimals instead of fractions. No tape diagrams are given; however, students may choose to draw diagrams if they find this helpful for matching the descriptions with the equations.

As students analyze different representations, they practice reasoning quantitatively and abstractly (MP2). In making connections across representations, they practice looking for and making use of structure (MP7).

The purpose of this discussion is to make connections between the numbers in the description and the numbers in the equation. Select 2–3 groups to share one of their sets of cards and how they matched the description with an equation. Discuss as many different sets of cards as the time allows.

Consider asking:

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”
• “How is the number in the equation related to the number in the description?”