# Lesson 15

Medidas de longitud

## Warm-up: Cuál es diferente: Medidas (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare length measurements given in different units, reminding about the relationships between yards, feet, and inches. In making comparisons, students need to attend to both the meaning of each unit and its relationships to other units.

### Launch

• Groups of 2
• Display the four measurements.
• “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 pareja cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

¿Cuál es diferente?

1. 3 pies

2. $$(3 \times 1)$$ yardas

3. $$\left(2 \times 18 \right)$$ pulgadas

4. $$\left(\frac{1}{3}+ \frac{1}{3}+ \frac{1}{3}\right)$$ yarda

### Activity Synthesis

• “Las cuatro cantidades son longitudes. ¿Qué observan acerca de estas longitudes?” // “All four quantities measure lengths. What do you notice about these lengths?” (They are in different units. Three of them are equivalent to 1 yard, and one of them is 3 yards.)
• “Si convertimos a pies las cantidades que son equivalentes a 1 yarda, ¿qué obtenemos?” // “If we convert the quantities that are equivalent to 1 yard into feet, what will they all be?” (3 feet)
• “¿Y si las convertimos a pulgadas?” // “What if we convert them into inches?” (36 inches)

## Activity 1: Lanzamientos de frisbee (15 minutes)

### Narrative

In this activity, students analyze length measurements, perform multiplication, and convert distances from yards to feet in order to compare and order them. The quantities in yards involve only whole numbers while those feet involve fractional amounts, to encourage students to convert from the larger unit to the smaller one.

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “¿Qué tuvieron en común las estrategias?” // “What did the strategies have in common?”, “¿En qué fueron diferentes?” // “How were they different?”, “¿Por qué al usar diferentes estrategias obtuvimos el mismo o los mismos resultados (o unos diferentes)?” // “Why did the different approaches lead to the same (or different) outcome(s)?”

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Consider asking students if they have played frisbee or another game that involves tossing an object. Alternatively, display a video clip of a frisbee game or a frisbee being tossed, or display a frisbee.
• Display the table. “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Read the opening paragraph and the bulleted information in the activity statement.
• “Averigüemos quiénes son los mejores lanzadores de frisbee de este grupo de amigos” // “Let’s find out who the top frisbee throwers are in this group of friends.”
• Display a yardstick and a foot-long ruler, if available.

### Activity

• 5 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who convert all the given distances into feet before finding the missing distances and ordering the measurements.

### Student Facing

Seis estudiantes lanzaban frisbees en el día de juegos al aire libre. Esta tabla muestra información sobre el primer lanzamiento de cada uno.

estudiante  distancia
Han  17 yardas
Lin  $$51\frac{1}{2}$$ pies
Clare  $$21 \frac{1}{3}$$ pies
Andre 22 yardas y 2 pies
Elena
Tyler
• El frisbee de Elena llegó 3 veces tan lejos como el de Clare.
• El frisbee de Andre llegó 4 veces tan lejos como el de Tyler.

1. Completa la tabla con las distancias de Elena y de Tyler. Explica o muestra cómo razonaste.

2. ¿Quiénes fueron los 3 mejores lanzadores en esa ronda?

Para averiguarlo, haz una lista de los estudiantes. Ordénalos según sus distancias en pies, de la más larga a la más corta.

puesto estudiante distancia (pies)
1
2
3
4
5
6

### Student Response

Students may be unsure how to find Elena’s throwing distance because it requires multiplying a mixed number by a whole number ($$3 \times21 \frac{1}{3}$$). Ask them if it’d help to consider the whole number (21) and the fraction ($$\frac{1}{3}$$) separately (and multiply them separately).

Students may rewrite $$21\frac{1}{3}$$ as $$\frac{64}{3}$$ and multiply it by 3 to get $$\frac{192}{3}$$, but may not know how this number compares to other distances because of its fraction form. Encourage them to think about what whole number this fraction approximates, or reason using smaller fractions with denominator 3. For instance, ask, “¿Cuánto es $$\frac{30}{3}$$?, ¿$$\frac{60}{3}$$?, ¿$$\frac{150}{3}$$?” // “How much is $$\frac{30}{3}$$? $$\frac{60}{3}$$? $$\frac{150}{3}$$?”

### Activity Synthesis

• Display the tables from the activity.
• Select students to complete the first table and share their reasoning.
• Then, select students to share their strategies for putting the distances in order. For each strategy, ask if others in the class reasoned the same way.
• “El lanzamiento de Tyler fue de 17 pies y el de Han fue de 17 yardas. ¿Podemos saber quién lanzó el frisbee más lejos sin convertir una unidad a la otra? Si es así, ¿cómo? Si no, ¿por qué no?” // “Tyler’s throw was 17 feet and Han’s was 17 yards. Can we tell who threw the frisbee farther without converting one unit to the other? If so, how? If not, why not?” (Yes. We know that 1 yard is greater than 1 foot, so 17 yards must be greater than 17 feet.)
• “¿La distancia de Han fue cuántas veces la distancia de Tyler?” // “How many times as far as Tyler’s distance was Han’s distance?” (3 times as far) “¿Cómo lo saben?” // “How do you know?” (A yard is 3 times as long as a foot, so 17 yards is 3 times as long as 17 feet.)

## Activity 2: Torres de piedras (20 minutes)

### Narrative

In this activity, students apply their knowledge of multiplicative comparison and ability to convert feet and inches to solve a logic puzzle. They use several given clues to determine the heights of four objects. As they use the clues to reason about the heights of the towers and who built them, students reason abstractly and quantitatively (MP2).

Action and Expression: Internalize Executive Functions. Provide students with access to sticky notes (of three different colors if available). Invite students to organize these sticky notes into a table, using the first color to record each person’s name, the second color to record clues about the height of each person’s tower, and the third color to record the actual height of each person’s tower. For students who need additional support, ask “¿Quién construyó la torre de piedra más baja? ¿Qué tan alta es esa torre?” // “Whose stone tower is the shortest? How tall is that tower?” and “¿Quién construyó una torre de 39 pulgadas de alto?” // “Whose tower is 39 inches tall?” Invite students to explore these questions using the process of elimination. Offer the sentence frame, “No puede ser ____ porque ____” // “It can’t be ____, because ____.”
Supports accessibility for: Conceptual Processing, Memory, Organization

### Launch

• Groups of 4

MLR5 Co-Craft Questions

• Display only the opening paragraph.
• “Escriban una lista de preguntas matemáticas que se podrían hacer acerca de esta situación” // “Write a list of mathematical questions that could be asked about this situation.”
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “¿Qué tienen en común estas preguntas? ¿En qué son diferentes?” // “What do these questions have in common? How are they different?”

### Activity

• “Completen el primer problema con su grupo. Después, trabajen individualmente en el último problema antes de discutirlo con su grupo” // “Work with your group to complete the first problem. Then, work on the last problem on your own before discussing it with your group.”
• 8–10 minutes: group work time
• 3–5 minutes: independent work time

### Student Facing

Mientras estaban en una excursión, un grupo de amigos hizo un concurso de apilar piedras para ver quién podía construir la torre más alta.

• La torre de piedras de Andre es 3 veces tan alta como la de Diego, pero Diego no construyó la torre más baja.
• La torre más alta mide 4 pies y 2 pulgadas de alto y pertenece a Tyler.
• Una persona construyó una torre que mide 39 pulgadas de alto.
• La torre de Tyler es 5 veces tan alta como la torre más baja.
1. ¿Qué tan alta es la torre de cada persona? Prepárate para explicar o mostrar cómo razonaste.

persona   altura de la torre (pulgadas)
Andre
Tyler
Clare
Diego
2. Elena se unió al grupo y construyó una torre que es 5 veces tan alta como la torre de Diego. ¿La torre de Elena mide más de 6 pies? Explica cómo razonaste.

### Student Response

Students may conclude that Andre’s tower is 117 inches (or $$39 \times 3$$), not realizing that this contradicts the second clue about Tyler having the tallest tower. Encourage students to check that their responses satisfy all the clues.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

Invite students to share how they reasoned about the height of each stone tower. Ask others if they reached the same conclusions but reasoned differently, or if they reached different conclusions.

“Una pista dice que la torre de Tyler es 5 veces tan alta como la torre más baja. Sabemos que la torre de Tyler mide 4 pies y 2 pulgadas. ¿Es más fácil encontrar la altura de la torre más baja usando 4 pies y 2 pulgadas o usando 50 pulgadas? ¿Por qué?” // “One clue says that Tyler’s tower is 5 times as tall as the shortest tower. We know that Tyler’s tower is 4 feet 2 inches. Is it easier to find the height of the shortest tower using 4 feet 2 inches or using 50 inches? Why?” (The first uses two different units, so we’d have to divide 4 feet by 5 and 2 inches by 5, or think about what number multiplied by 5 gives 4 and 2. If we use inches, we’re dealing with one number that is clearly a multiple of 5.)

“¿Cómo decidieron si la torre de Elena es más alta o más baja que 6 pies? ¿Convirtieron los 6 pies a pulgadas, convirtieron la altura de la torre de Diego a pies o lo hicieron de otra forma?” // “How did you decide whether Elena’s tower is greater than 6 feet? Did you convert the 6 feet into inches, convert Diego’s tower into feet, or another way?”

Record strategies. Highlight that while it is often helpful to express a larger unit in terms of a small unit, some problems can be reasoned without doing so.