# Lesson 4

Exploremos relaciones entre valores posicionales (optional)

## Warm-up: Observa y pregúntate: Mantén el equilibrio (10 minutes)

### Narrative

The purpose of this warm-up is for students to recognize that a balance shows when two weights are equal. This will be useful when students look at weights representing the different decimal place values, providing students with a concrete way to think about the meaning of the digits in a decimal number and the multiplicative relationships between the place values (MP7). The weights will also be used to connect the word form of a decimal number to thousandths and the decimal form.

### Launch

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

### Student Facing

¿Qué observas? ¿Qué te preguntas?

### Activity Synthesis

• “Cada uno de los pesos pequeños pesa 0.001 de una onza. El peso grande pesa una centésima de una onza. Si en un lado hubiera 20 pesos y cada uno pesara 0.001 de una onza, ¿qué debería haber en el otro lado para mantener el equilibrio?” // “Each of the smaller weights weigh 0.001 of an ounce. The larger weight weighs one hundredth of an ounce. If there were 20 weights that each weighed 0.001 of an ounce on one side, what would need to be on the other side to stay balanced?” (2 hundredths, 1 hundredth and 10 thousandths, 20 thousandths)

## Activity 1: Equilibremos el peso (15 minutes)

### Narrative

The purpose of this activity is for students to examine relationships between the different decimal place values. In earlier lessons, students represented decimal numbers using words, fractions, diagrams, and symbols. The diagrams help to reveal that a thousandth is 1 tenth of a hundredth and a hundredth is 1 tenth of a tenth. In this activity, students systematically examine these relationships. For example, there are many different ways to represent 2 tenths. It is also 20 hundredths or 200 thousandths or 1 tenth and 10 hundredths. Through the idea of weights, students investigate these different equivalences. The weights give students a visual and physical way to reason about the different place values and their relationships (MP2, MP7).

The activity synthesis focuses on two key ways to balance a weight or represent a decimal number:

• use the least number of weights to represent a three-digit decimal weight, which is the expanded form students studied in the previous lesson (for example 3 tenths, 8 hundredths, and 5 thousandths).
• use only the smallest weights, thousandths, to represent a three-digit weight, which is the way the decimal number is usually said in words or written as an equivalent fraction (385 thousandths).
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 “¿Qué colecciones de pesos equilibrarán las pepitas?” // “What sets of weights will balance the nuggets?”. 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.

### Launch

• Display the image of balance.
• “Las balanzas se usan para pesar cosas. En un lado se pone el objeto que se quiere pesar y en el otro lado se ponen pesos. Cuando los dos lados se equilibran es porque tienen el mismo peso” // “Balances are used to weigh things. On one side you put the object you want to weigh and on the other side you put weights. When the two sides balance, they have the same weight.”

### Activity

• 8–10 minutes: independent work time
• Monitor for students who explicitly use the values of the different decimal places in their reasoning by:
• thinking about the digits in the decimals separately
• understanding how to express tenths in terms of hundredths or thousandths, and hundredths in terms of thousandths

### Student Facing

En cada problema tienes una balanza y pesos de 0.1 onzas, 0.01 onzas y 0.001 onzas.

1. Una pepita de oro pesa 0.2 onzas.
1. ¿Qué colección de pesos podrías usar para equilibrar la pepita? Explica o muestra tu razonamiento.
2. ¿Qué otra colección de pesos podrías usar para equilibrar la pepita? Explica o muestra tu razonamiento.
3. ¿Cuántos pesos de 0.01 onzas necesitarías para equilibrar la pepita?, ¿cuántos pesos de 0.001 onzas?
2. Otra pepita pesa 0.385 onzas.
1. ¿Qué colección de pesos podrías usar para equilibrar la pepita? Explica o muestra tu razonamiento.
2. ¿Cuál es el menor número de pesos que puedes usar para equilibrar la pepita? Explica o muestra tu razonamiento.
3. ¿Cuál es el mayor número de pesos que puedes usar para equilibrar la pepita? Explica o muestra tu razonamiento.
3. Escribe un número decimal que represente el peso de las pepitas de oro que se equilibran con:
1. 266 pesos de 0.001 onzas
2. 150 pesos de 0.01 onzas
3. 27 pesos de 0.1 onzas

### Activity Synthesis

• Invite students to share their responses for the weights they use to balance the 0.385 ounce gold nugget.
• “¿Cómo saben que 3 pesos de 0.1 onzas, 8 pesos de 0.01 onzas y 5 pesos de 0.001 onzas van a funcionar?” // “How do you know that 3 of the 0.1 ounce weights, 8 of the 0.01 ounce weights, and 5 of the 0.001 ounce weights will work?” (The 3 tenth ounce weights give the 3 from the decimal, the 8 hundredth ounce weights give the 8, and the 5 thousandth ounce weights give the 5.)
• “Podemos representar este número decimal en su forma desarrollada” // “We can represent this with expanded form.”
• Display equation: $$0.385 = 3 \times 0.1 + 8 \times 0.01 + 5 \times 0.001$$
• “¿Cómo saben que 385 pesos de 0.001 onzas también van a funcionar?” // “How do you know that 385 of the 0.001 ounce weights will also work?” (Because that is the same as 3 tenth ounce weights, 8 hundredth ounce weights, and 5 thousandth ounce weights.)
• Refer to the decimal, 0.385
• “¿Cómo pueden decir este número decimal?” // “How can you say or name this decimal number?” (Three hundred eighty five thousandths or 3 tenths, 8 hundredths, and 5 thousandths)

## Activity 2: Pesos y valores posicionales (10 minutes)

### Narrative

The purpose of this activity is to provide further practice relating the different forms of decimals. This includes expanded form, word form, and decimal form. Using the balance and weight of gold nuggets as a context, students go back and forth between different ways of representing these weights (MP2). The synthesis highlights the meaning of the digits in a decimal and how that relates to the expanded form of the decimal.

### Launch

• Display the image.
• “¿Qué número decimal puedo escribir para representar el peso de las pepitas de oro?” // “What decimal can I write for the weight of the gold nugget?” (0.124)
• Write equation: $$0.124 = \left(1 \times 0.1\right) + \left(2 \times 0.01\right) + \left(4 \times 0.001\right)$$
• “¿Cómo se ve la forma desarrollada de 0.124 en la balanza?” // “How does the balance show the expanded form of 0.124?” (There is 1 tenth of an ounce weight, 2 hundredth of an ounce weights, and 4 thousandth of an ounce weights.)

### Activity

• 8–10 minutes: independent work time
• Monitor for students who see, in the first problem, that the digits in the decimal for the gold weights are the same as the number of weights for that decimal place value.

### Student Facing

1. Para equilibrar algunas pepitas de oro se usan varios pesos. Escribe el peso de cada pepita de oro en forma desarrollada.
1. tres pesos de 0.1 onzas, cinco pesos de 0.01 onzas y ocho pesos de 0.001 onzas
2. seis pesos de 0.1 onzas y dos pesos de 0.001 onzas
3. dos pesos de 0.01 onzas y seis pesos de 0.1 onzas
2. Estos son los pesos, en palabras, de algunas pepitas de oro. Escribe los pesos en forma desarrollada.
1. doscientas ochenta y tres milésimas de una onza
2. cuatrocientas nueve milésimas de una onza
3. Una pepita de oro pesa 0.527 onzas.
1. ¿Cuál es el valor de cada uno de los dígitos del decimal 0.527?
2. ¿Cómo se ve el valor de cada dígito del número decimal 0.527 en su forma desarrollada?

### Activity Synthesis

• Invite students to share the expanded form of the decimal 0.527.
• Display the expression: $$\left(5 \times 0.1 \right) +\left(2 \times 0.01\right) +\left( 7 \times 0.001\right)$$
• “¿Cuál es el valor del 5 en 0.527?” // “What is the value of the 5 in 0.527?” (5 tenths)
• “¿Cómo nos muestra esto la forma desarrollada?” // “How does the expanded form show this?” (It shows the 5 is $$5 \times 0.1$$ or 5 tenths.)
• “¿Cuál es el valor del 7 en 0.527?” // “What is the value of the 7 in 0.527?” (7 thousandths)
• “¿Cómo nos muestra esto la forma desarrollada?” // “How does the expanded form show this?” (It shows the 7 is $$7 \times 0.001$$ or 7 thousandths.)
• “¿Qué diferencias hay entre pasar de la forma en palabras a la forma desarrollada y pasar de la forma decimal a la forma desarrollada?” // “How is going from word form to expanded form different than going from decimal form to expanded form?” (The decimal form shows the place values. With the word form, everything is given in terms of thousandths, so I need to figure out what the individual place values of the number are.)

## Activity 3: Comparemos valores posicionales usando pesos (10 minutes)

### Narrative

The purpose of this activity is for students to use the weights from the previous activity to support place value understanding, specifically to see the multiplicative relationships between different decimal place values (MP7). These relationships will be taken up in greater detail in the next unit but the weights provide a convenient way to see these relationships which complements the diagrams students used in earlier lessons.

Students first compare weights of two gold nuggets, one weighed using 0.1 ounce weights and the other using 0.01 ounce weights. The two nuggets have the same weight because ten 0.01 ounce weights are equivalent to one 0.1 ounce weight, as students saw in the warm-up. Students move from here to making multiplicative comparisons between place values. They can use the weights to help visualize or calculate or they might use a diagram like those in the previous lesson.

Representation: Internalize Comprehension. Synthesis: Provide students with a partially completed graphic organizer, such as a two column table, to record the relationship between tenths and hundredths, hundredths and thousandths, and tenths and thousandths.
Supports accessibility for: Conceptual Processing, Memory

• Groups of 2

### Activity

• 2 minutes: independent work time
• 8 minutes: partner work time
• Monitor for students who use different strategies to compare the values of the 6s in the gold nugget weights:
• using the value of each place in the decimal
• thinking about fractions or representing the decimals as fractions
• using a diagram such as a hundredths grid

### Student Facing

1. ¿Cuántos pesos de 0.01 onzas equilibran un peso de 0.1 onzas? Explica o muestra tu razonamiento.
2. ¿Cuántos pesos de 0.001 onzas equilibran un peso de 0.1 onzas? Explica o muestra tu razonamiento.
3. La tabla muestra los pesos de 3 de las pepitas de oro que Diego y sus amigos encontraron lavando oro.

Llena los espacios en blanco. Explica o muestra tu razonamiento.

oro peso
(gramos)
pepita A 0.6
pepita B 0.06
pepita C 0.006
1. La pepita A pesa $$\underline{\hspace{0.9cm}}$$ veces lo que pesa la pepita B.
2. La pepita A pesa $$\underline{\hspace{0.9cm}}$$ veces lo que pesa la pepita C.
3. La pepita C pesa $$\underline{\hspace{0.9cm}}$$ veces lo que pesa la pepita B.
4. La pepita C pesa $$\underline{\hspace{0.9cm}}$$ veces lo que pesa la pepita A.

### Activity Synthesis

• “¿En qué se parecen los pesos de las pepitas? ¿En qué son diferentes?” // “How are the weights of the nuggets the same? How are they different?” (They all have a 6 in them. They each balance with 6 of one of the unit weights. The value of the 6 is different for each nugget.)
• “Comparen las centésimas (usadas para el peso de la pepita B) y las décimas (usadas para el peso de la pepita A). ¿Qué pueden decir?” // “How do hundredths, the weights for B, compare to tenths, the weights for A?” (There are 10 hundredths in each tenth or a tenth of a tenth is a hundredth.)
• “¿La pepita B pesa cuántas veces lo que pesa la pepita A? ¿Cómo lo saben?” // “Nugget B weighs how many times as much as Nugget A? How do you know?” ($$\frac{1}{10}$$ because Nugget A is 6 tenths and a tenth of that is 6 hundredths since a tenth of a tenth is a hundredth.)
• “Comparen las milésimas (usadas para el peso de la pepita C) y las décimas (usadas para el peso de la pepita A). ¿Qué pueden decir?” // “How do thousandths, the weights for C, compare to tenths, the weights for A?” (There are 100 thousandths in a tenth and one hundredth of a tenth is a thousandth.)
• “¿La pepita A pesa cuántas veces lo que pesa la pepita C?” // “Nugget A weighs how many times as much as Nugget C?” (100, there are 100 thousandths in a tenth.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy investigamos distintas formas de escribir números decimales al pensar en una balanza y en las maneras en las que podemos equilibrar un objeto dado. Estos son los pesos que equilibran dos pepitas de oro” // “Today we investigated different ways to write decimal numbers by thinking about a balance and the different ways we can balance a given object. Here are the weights that balance two gold nuggets.”

Display the table:

oro peso
pepita 1 dos pesos de 0.1 onzas
pepita 2 veinte pesos de 0.01 onzas

//

gold weight
nugget 1 two 0.1 ounce weights
nugget 2 twenty 0.01 ounce weights
“¿Cuántas onzas pesan las dos pepitas de oro?” // “How many ounces do the two gold nuggets weigh?” (2 tenths and 20 hundredths)

“¿Cómo saben que esos dos pesos son equivalentes?” // “How do you know those two weights are equivalent?” (1 tenth is 10 hundredths so 2 tenths is 20 hundredths)

“¿Cuántos pesos de milésima de onza se necesitarían para equilibrar cada una de estas pepitas?” // “How many thousandth ounce weights would you need to balance each of these nuggets?” (I would need 200 thousandth weights because 10 of them is a hundredth and so 20 hundredths is 200 thousandths.)

Display the decimals: 0.2, 0.20, 0.200

“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (I notice that they all have the same value, 2 tenths, 20 hundredths, 200 thousandths. I wonder why there are so many different ways to express a decimal number.)