# Lesson 9

Ordenemos números decimales

### Narrative

The purpose of this True or False is for students to demonstrate strategies and understandings they have for comparing decimals. These understandings will be valuable when students order decimals later in this lesson. As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).

### Launch

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

En cada caso, decide si la afirmación es verdadera o falsa. Prepárate para explicar tu razonamiento.

• $$0.909 > 0.91$$
• $$4.1 < 4.100$$
• $$0.99 < 0.999$$

### Activity Synthesis

• “¿La afirmación $$0.909 > 0.91$$ es verdadera o falsa? ¿Cómo lo saben?” // “Is the statement $$0.909 > 0.91$$ true or false? How do you know?” (It is false because 0.909 has 9 tenths and 9 thousandths and 0.91 is 9 tenths and 1 hundredth. 1 hundredth is greater than 9 thousandths.)
• Display: 0.909 and 0.910
• “¿Cómo nos ayuda escribir así los números cuando los comparamos?” // “How does writing the numbers like this help to compare them?” (I can see that 0.910 has 10 thousandths compared to 9 for 0.909.)

## Activity 1: Atrapado en el medio (20 minutes)

### Narrative

The purpose of this activity is for students apply what they have learned about comparing decimals to find numbers that lie between two other decimal numbers. Students may draw number line diagrams, if it helps them, or they may use their understanding of place value.

In each case, there are many different decimal numbers between the two and this will be brought out in the activity synthesis. The last question in this activity is exploratory. Students may say that there is no number between 1.731 and 1.732 or they may say that it looks like there is and they cannot name it yet. The important observation is that the number line suggests that there are numbers in between but we cannot name any of those numbers yet. This question gives students an opportunity to make sense of a problem and some students may propose an answer, using fractions for example (MP1).

Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Begin with more accessible values. For example, allow students to complete one side of the statement at a time.
Supports accessibility for: Conceptual Processing, Attention

• Groups of 2

### Activity

• 10 minutes: independent work
• 5 minutes: partner discussion
• Monitor for students who:
• use a number line
• count up by hundredths or thousandths between the intervals to find a number in the middle
• use place value understanding to introduce new places in the decimals when needed

### Student Facing

1. Llena cada espacio para que la afirmación sea verdadera. Prepárate para explicar tu razonamiento. Usa las rectas numéricas si crees que te pueden ayudar.

1. $$786.2 <\, \underline{\hspace{2cm}}\, < 786.3$$

2. $$9.99 < \,\underline{\hspace{2cm}}\, < 10$$

3. $$0.46 > \,\underline{\hspace{2cm}} \,>0.45$$

4. $$0.5 < \,\underline{\hspace{2cm}}\, < 0.51$$

5. $$0.99 < \,\underline{\hspace{2cm}} \,< 0.999$$

2. Kiran dice que no hay números entre 1.731 y 1.732. ¿Estás de acuerdo con Kiran? Si te ayuda, usa la recta numérica.

### Activity Synthesis

• Display the inequality: $$0.99 < \underline{\hspace{2cm}} < 0.999$$
• “¿Qué números pueden hacer que esta afirmación sea verdadera?” // “What are some possible numbers that will make this true?” (0.995, 0.991, 0.997)
• “¿Qué observan acerca de todos los números posibles?” // “What do you notice about all the possible numbers?” (They all have 9 tenths and 9 hundredths and also some thousandths.)
• “¿Por qué esto tiene sentido?” // “Why does this make sense?” (They need 9 tenths and 9 hundredths to be as big as 0.99 and some thousandths to be bigger. There can be at most 8 thousandths so the number will be less than 0.999.)
• Display number line from the student solution or use a student generated image.
• “¿Creen que existen números entre 1.731 y 1.732?” // “Do you think there are numbers between 1.731 and 1.732?” (It looks like there are lots of them but we don’t know what any of those numbers are.)

## Activity 2: De menor a mayor (15 minutes)

### Narrative

The purpose of this activity is for students to apply what they have learned about place value and decimals to order several decimals from least to greatest. Students may draw number line diagrams, if it helps them, but will need to think strategically about the endpoints that they choose if they want all 3 numbers to fit. They can also order the numbers by looking carefully at place value to compare pairs of decimals (MP7).

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to ordering the sets of numbers from least to greatest. Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.

• Groups of 2

### Activity

• 8 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who:
• use their understanding of place value to compare the numbers
• use a number line to visualize how the numbers compare

### Student Facing

1. Escribe cada conjunto de números en orden de menor a mayor.

1. 67.020, 67.200, 67.002
2. 1.101, 1.02, 1.1
3. 0.333, 0.323, 0.3
4. 99.99, 99.09, 99.091

### Activity Synthesis

• Display the numbers: 1.101, 1.02, 1.1
• “¿Cómo decidieron cuál de estos números es el menor?” // “How did you decide which of these numbers is the smallest?” (They all have 1 and some more. The second one only has hundredths while the other two have tenths, so 1.02 is the smallest.)
• “¿Cómo decidieron cuál de estos números es el mayor?” // “How did you decide which one of these numbers is the greatest?” (1.101 has a tenth and a thousandth while 1.1 only has a tenth, so 1.101 is the greatest.)
• Use a student generated number line or display the number line from student solutions and consider asking:
• “¿Cómo saben que 99.091 está entre 99.09 y 99.99 en la recta numérica?” // “How do you know that 99.091 is between 99.09 and 99.99 on the number line?” (It has a thousandth more than 99.09 and has no tenths so is smaller than 99.1 and definitely smaller than 99.99.)
• “¿Por qué es difícil ubicar 99.091 con precisión en la recta numérica?” // “Why is it hard to locate 99.091 precisely on the number line?” (It is really close to 99.09, just one thousandth to the right.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy ordenamos números decimales” // “Today we ordered decimals.”

“Describan los pasos que usarían para ordenar un conjunto de números de menor a mayor” // “Describe the steps you would use to put a set of numbers in order from least to greatest.” (Start with the digits in the largest place value and compare them. When they are the same, compare the digits in the next largest place. If all the digits are the same, then the numbers are the same. Wherever they differ first, the number with the larger digit in that place is larger.)