# Lesson 3

Decimales en rectas numéricas

## Warm-up: Cuál es diferente: Decimales y fracciones (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare different representations of numbers. In making comparisons, they solidify their understanding of the connections across representations.

### Launch

• 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

### Activity

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

### Student Facing

¿Cuál es diferente?

### Activity Synthesis

• “¿Cómo podríamos ajustar una o más opciones para que todas representen el mismo valor?” // “How might we revise one or more of the options so that they all represent the same value?” (Sample responses:
• Change the $$\frac{80}{10}$$ to $$\frac{8}{10}$$ or $$\frac{80}{100}$$.
• Change option A to say “eight,” option C to say “8” or “8.0,” and the label for the first tick mark in option D to say “1.”)

## Activity 1: Puntos en rectas numéricas (20 minutes)

### Narrative

In this activity, students reason about the relative size of decimals by locating them on a number line. As in a previous activity, they rely on their experience of locating fractions on a number line and the relationship of the decimal values relative to 0 and 1.

If desired and logistically feasible, consider carrying out the activity on a giant number line rather than on paper.

• Stretch a long strip of tape across a wall, at least 8 or 10 feet long. Partition the tape into 12 equal intervals, using shorter pieces of tape as tick marks. Label the locations of 0 and 1.
• Give each group of students 2–3 dot stickers, 4–5 sticky notes, and a thick marker.
• Ask each group to write (on sticky notes) labels for two of the tick marks, with one sticky note for each label.
• Assign each group one or two of the eight decimals in the second problem. Ask them to locate their assigned decimals on the number line, using dot stickers to mark the location and sticky notes to label them.
Action and Expression: Internalize Executive Functions. Synthesis: Invite students to plan a strategy, including the tools they will use, for the first two steps of the activity. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Organization, Social-Emotional Functioning

### Launch

• Groups of 3–4

### Activity

• If creating a giant number line, lead the activity as outlined in the Activity Narrative. Otherwise, ask students to work with their group on the first two problems.
• Pause and discuss:
• how students knew where to put each decimal
• how the number line could help us see the least and greatest
• “Tómense unos minutos para completar en silencio el resto de la actividad” // “Take a few quiet minutes to complete the rest of the activity.”
• 5–6 minutes: independent work time

### Student Facing

1. Escribe debajo de cada marca el número que la marca representa.

2. Estos son ocho números.

• 0.10

• 0.40

• 0.80

• 1.10

• 0.15

• 0.45

• 0.75

• 1.05

1. Ubica y marca cada número en la recta numérica.
2. ¿Cuál es el mayor número? ¿Cuál es el menor? Explica cómo puede ayudar la recta numérica a encontrar el mayor número y el menor número.
3. Ubica y marca estos números en la recta numérica.

• 0.24

• 0.96

• 0.61

• 1.12

• 0.08

4. Haz que cada afirmación de comparación sea verdadera completándola con dos números de la pregunta anterior.

1. ________ es mayor que _______.

2. ________ es menor que _______.

3. ________ es el mayor número.

### Student Response

Students may be unsure whether to label the tick marks between 0 and 1 in terms of tenths or hundredths, or whether to use fraction or decimal notation. Consider asking: “¿Qué marcas podemos poner para ayudarnos a ubicar los decimales de la actividad?” // “What labels might be helpful for locating the decimals in the activity?”

### Activity Synthesis

• Select students to display their completed number lines (showing the locations of five additional decimals).
• Invite the class to agree or disagree with the locations of the point and discuss any disagreements.
• Select other students to share their comparison statements and how they know each statement is true.

## Activity 2: Comparemos decimales (15 minutes)

### Narrative

In this activity, students continue to compare decimals to hundredths. They begin by reasoning with a number line and work toward generalizing their observations. Some students may compare two numbers by analyzing the value of the digits in the same place (for example, the tenths in one number and the tenths in the other), but comparing decimals by place value is a standard for grade 5 and thus not expected at this point.

MLR8 Discussion Supports. Display sentence frames to support partner discussion: “Observé _____, entonces yo . . . ” // “I noticed _____ so I . . .” and “Estoy de acuerdo / en desacuerdo porque . . .” // “I agree/disagree because . . . .”

• Groups of 2

### Activity

• “En silencio, trabajen unos minutos en la actividad. Luego, compartan con su compañero cómo pensaron” // “Take a few quiet minutes to work on the activity. Then, share your thinking with your partner.”
• 7–8 minutes: independent work time
• 3–4 minutes: partner discussion
• For the comparisons in the second problem, monitor for students who:
• name the decimal in words and compare the number of hundredths (for instance, 62 hundredths and 26 hundredths)
• relate the decimals to benchmarks such as 0, 0.5, and 1

### Student Facing

1. Esta recta numérica tiene dos puntos marcados.

1. Escribe el decimal que está en el punto A.
2. ¿El decimal del punto A es menor que o mayor que 0.50? Explica o muestra tu razonamiento.

3. ¿El decimal del punto B es mayor que o menor que 0.06? Explica tu razonamiento.
4. Estima el decimal del punto B.

2. En cada caso, compara los números escribiendo un $$<$$, un $$>$$, o un $$=$$. ¿Se te ocurre alguna forma de comparar sin usar una recta numérica? Prepárate para explicar tu razonamiento.

1. $$0.51 \underline{\hspace{0.5in}} 0.09$$

2. $$0.19 \underline{\hspace{0.5in}} 0.91$$

3. $$0.45 \underline{\hspace{0.5in}} 0.54$$

4. $$0.62 \underline{\hspace{0.5in}} 0.26$$

5. $$1.02 \underline{\hspace{0.5in}} 0.95$$

6. $$0.3 \underline{\hspace{0.5in}} 0.30$$

7. $$4.01 \underline{\hspace{0.5in}} 4.10$$

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Hoy comparamos decimales teniendo en cuenta sus décimas y sus centésimas” // “Today we compared decimals in tenths and hundredths.”

“¿Cómo podemos usar una recta numérica para ayudarnos a hacer comparaciones?” // “How can we use a number line to help us make comparisons?” (We can plot the decimals on the number line. The one farther to the right is the greater decimal.)

“¿Cómo podríamos comparar decimales sin usar una recta numérica? ¿Qué estrategias usaron para completar las afirmaciones de comparación de la última actividad?” // “How might we compare decimals without using a number line? What strategies did you use when completing the comparison statements in the last activity?”

If not mentioned in students’ explanations, highlight the following reasoning strategies:

• Name the decimals in words and compare the number of hundredths. (For instance, 51 hundredths is more than 9 hundredths.)
• Compare each decimal to benchmarks like 0, 0.5, 1, or other decimals. (For instance, 0.51 is close to 0.50, while 0.09 is close to 0.10 or close to 0.)