Lesson 3

This warm-up prompts students to compare four expressions. The particular expressions chosen give students a chance to focus on several important features, including:

• the operations
• the values of the expressions
• the types of numbers in the expressions (whole numbers versus decimals)

Students work in this lesson to express decimals in many different forms, and this warm-up gives students some familiarity thinking about some of those different forms. 

• Groups of 2
• Display the image.
• “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• Display the expression: \(2 \times 0.1 + 6 \times 0.01\)
• “¿En qué se diferencia esta expresión de las demás?” // “How is this expression different from the others?” (It is written as a sum. The different place values are written separately.)
• “Hoy vamos a representar números decimales de esta manera” // “Today we are going to represent decimal numbers in this way.”

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 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 is shown in expanded form by writing the \(0.8\) as \(8 \times 0.1\). 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. 

• Groups of 2

• 15 minutes: partner work time

• Display the diagram from the student workbook that shows \(0.835\).
• Consider circling each part of the diagram as students respond to the questions.
• “¿Qué parte del diagrama muestra \(8 \times 0.1\)? ¿Cómo lo saben?” // “What part of the diagram shows \(8 \times 0.1\)? How do you know?” (The 8 shaded horizontal strips because they are each tenths or 0.1.)
• “¿Qué parte del diagrama muestra \(3 \times 0.01\)? ¿Cómo lo saben?” // “What part of the diagram shows \(3 \times 0.01\)? How do you know?” (The 3 single shaded squares because they are hundredths or 0.01.)
• “¿Qué parte del diagrama muestra \(5 \times 0.001\)? ¿Cómo lo saben?” // “What part of the diagram shows \(5 \times 0.001\)? How do you know?” (The 5 tiny shaded pieces because they are each thousandths or 0.001.)
• “¿Qué representa el 5 en 0.105?” // “What does the 5 in 0.105 represent?” (5 thousandths)
• “Puedo escribir 0.105 como \(1 \times 0.1 + 5 \times 0.001\) para mostrar que es 1 décima y 5 milésimas. Esto se llama la ‘forma desarrollada’ de un número decimal” // “I can write 0.105 as \(1 \times 0.1 + 5 \times 0.001\) to show that it is 1 tenth and 5 thousandths. This is called the expanded form of a decimal.”

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).

• Groups of 2
• Display the first image that represents the number 0.742 in the student workbook.
• “¿De qué maneras podemos representar el número que se muestra en el diagrama?” // “What are some different ways we can represent the number shown in the diagram?”
  • 0.742
  • seven hundred forty-two thousandths
  • \(0.7 + 0.04 + 0.002\)
  • \((7\times 0.1)+(4 \times 0.01)+(2\times 0.001)\)
  • \(\frac {742}{1,000}\)
• 1 minute: quiet think time
• Share and record responses.

• “Ahora representen cada número de tantas maneras como puedan” // “Now find as many ways as you can to represent each number.”
• 2 minutes: independent work time
• 6 minutes: partner work time

If students are not able to find multiple ways to represent a given number, refer to the list from the launch of the activity.

• Invite students to share their representations of \(\left(3 \times 0.1\right) + \left(6 \times 0.01\right) + \left(8 \times 0.001\right)\).
• Display: 0.368
• “¿Cómo se relaciona cada dígito del número decimal con la forma desarrollada?” // “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: \(\frac{368}{1,\!000}\)
• “¿Cómo se relaciona la forma desarrollada con la fracción?” // “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.
• “¿Cómo se relaciona la forma desarrollada con el diagrama?” // “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.)