In a previous course, students multiplied a decimal fraction by a whole number. In previous lessons, students wrote decimal fractions in decimal form. The purpose of this activity is for students to use the expanded form of a decimal number to the thousandth. Students relate expanded form to both diagrams and decimal numbers. The expanded form of a decimal number highlights the value of each digit. For the number 0.835, for example, the 8 represents 8 tenths. This decimal is shown in expanded form by writing the as . Students practice relating decimals, diagrams, and expanded form and then are formally introduced to the term expanded form, as it applies to decimals, in the Activity Synthesis (MP2). The notation of expanded form is a generalization of what students saw in a previous grade with whole numbers. 

This activity uses MLR3 Critique, Correct, Clarify. Advances: reading, writing, representing

MLR3 Critique, Correct, Clarify

• Display Mai's response for students to consider.
• Read the explanation aloud.
• “What parts of this response are unclear, incorrect, or incomplete?” ( doesn't match any parts of 0.105. It's missing an expression that matches 5 thousandths.)
• 1 minute: quiet think time
• 2 minutes: partner discussion
• Invite 2–3 groups share what they discussed. Record for all to see.
• “With your partner, work together to write a revised equation and explain how it represents 0.105.”
• Display and review the following criteria: 
  • Write an expression using expanded form.
  • Write an explanation for each part of the expression.
  • Use “tenths,” “hundredths,” and “thousandths,”
• 3–5 minutes: partner work time
• Select 1–2 groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each group shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
• “What is the same and different about the explanations?” (Mai used the same whole numbers. Both expressions have more than 1 other expression in them. Mai's expression represented 0.15.)

The purpose of this activity is for students to practice different ways of expressing decimal numbers to the thousandth. In addition to standard decimal digit form, these ways include:

• Diagrams
• Expanded form
• Fractions
• Words

The goal of the Activity Synthesis is to show how the different ways to represent a decimal are interrelated. This gives students an opportunity to make sense of each form and how it relates to the others (MP2).

• Invite students to share their representations of .
• Display: 0.368
• “How does each digit in the decimal relate to the expanded form?” (The 3 is 3 groups of 1 tenth or 0.3, the 6 is six groups of 1 hundredths or 0.06, and the 8 is 8 groups of 1 thousandth or 0.008.)
• Display:
• “How does the expanded form relate to the fraction?” (The 300 is the 3 tenths, the 60 is the 6 hundredths, and the 8 is the 8 thousandths.)
• Display student work that shows 0.368 represented on the hundredths grid.
• “How does the expanded form relate to the diagram?” (The 3 tenths are the top 3 rows. The 6 hundredths are the 6 squares in the next row. The 8 thousandths are the small rectangles.)