Lesson 17

Más problemas sobre perímetros

Warm-up: Verdadero o falso: Fracciones y números enteros (10 minutes)

Narrative

This True or False routine prompts students to recall what they know about addition of fractions with a common denominator and about fractions that are equivalent to whole numbers. The understandings elicited here will be helpful later in the lesson when students find sums or differences of two fractions, or of a whole number and a fraction, to solve problems about the perimeter of rectangles with fractional side lengths.

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 strategies.
• Repeat with each statement.

Student Facing

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

• $$\frac {8}{12} + \frac{3}{12} + \frac{9}{12} + \frac{4}{12} = 2$$
• $$\frac{20}{4} + \frac{10}{4} + \frac{6}{4} = 8$$
• $$2 = \frac{59}{100} + \frac{41}{100} + \frac{89}{100} + \frac{11}{100}$$
• $$2 = \frac{3}{8} + \frac{3}{8} + \frac{12}{8}$$

Activity Synthesis

• “En cada ecuación, ¿cómo sabemos que la suma de las fracciones se puede escribir como un número entero? Por ejemplo, ¿cómo sabemos que $$\frac{24}{12}$$ y $$\frac{36}{4}$$ se pueden escribir como números enteros?” // “In each equation, how do we know that the sum of the fractions can be written as a whole number? For example, how do we know that $$\frac{24}{12}$$ and $$\frac{36}{4}$$ can each be written as a whole number?” (There are 2 groups of $$\frac{12}{12}$$ in $$\frac{24}{12}$$ and 9 groups of $$\frac{4}{4}$$ in $$\frac{36}{4}$$.)
• “¿Cómo sabemos si una fracción es equivalente a un número entero?” // “How do we know if any fraction is equivalent to a whole number?” (The numerator is a multiple of the denominator.)

Activity 1: Por las paredes a pasos diminutos (15 minutes)

Narrative

In this activity, students use their knowledge of feet and inches and the perimeter of a rectangle to solve problems in context. Given the perimeter and one length of a room, they determine its width in feet and the distance along two walls in inches. Students could reason about each problem in a number of ways. As they interpret the quantities in the situation and represent them with expressions or equations, students practice reasoning quantitatively and abstractly (MP2).

MLR7 Compare and Connect. Synthesis: Invite students to prepare a visual display that shows the strategy they used to determine how many inches the ant traveled. Encourage students to include details that will help others interpret their thinking. Give students time to investigate each others’ work. During the whole-class discussion, ask students, “¿Alguien resolvió el problema de la misma manera, pero lo explicaría de otra forma?” // “Did anyone solve the problem the same way, but would explain it differently?”, “¿Qué tuvieron en común las estrategias?” // “What did the strategies have in common?”, “¿En qué fueron diferentes?” // “How were they different?”

Launch

• Groups of 2
• “Resolvamos algunos problemas sobre las longitudes de los lados y el perímetro de una habitación” // “Let’s solve some problems about the side lengths and perimeter of a room.”

Activity

• 6–8 minutes: independent work time
• 4–5 minutes: partner discussion
• Monitor for the different ways students reason about the number of inches traveled by the ant in the last problem. They may:
• convert the perimeter from feet to inches and divide by 2
• convert the length and the width into inches separately and then add them
• find the sum of the length and width in feet and then convert that sum into inches

Student Facing

Una habitación rectangular tiene un perímetro de 39 pies y un largo de $$10\frac{1}{2}$$ pies.

1. ¿Cuál es el ancho de la habitación, en pies? Explica o muestra cómo razonaste.
2. Una hormiga se paró en una esquina. Caminó a lo largo de una pared, en línea recta. Luego caminó a lo largo de la siguiente pared, también en línea recta, hasta terminar en la esquina opuesta a la esquina donde comenzó. ¿Cuántas pulgadas recorrió? Explica o muestra cómo razonaste.

Student Response

To find the width of the room, students may subtract $$10\frac{1}{2}$$ feet from 39 feet and not remember that there are two sides that are $$10\frac{1}{2}$$ feet long and there are two sides with the same width. Encourage them to draw a diagram of the room and to label the sides with known measurements.

Activity Synthesis

• Invite students to briefly share their response to the first problem and then focus the discussion on the last problem.
• Select students with different strategies to find the distance the ant walked to share their reasoning.

Activity 2: Las medidas desconocidas (20 minutes)

Narrative

This activity allows students to consolidate their learning from the past few units to solve problems about length measurements in a mathematical context.

First, students find the perimeter or missing side length of various quadrilaterals. To do so, they apply what they know about adding and subtracting fractions and about rewriting certain fractions as whole numbers. Next, they determine which pairs of figures have a certain multiplicative relationship—for instance, which figure has a perimeter that is 9 times that of another figure. Because the measurements are in different units, students need to attend to the relationship between units and perform conversions accordingly.

The activity can be done in the format of a gallery walk or by giving each group a full set of the diagrams from the blackline master.

Here are the images on the blackline master for reference:

Perimeter =  9 km

Perimeter = __________

Perimeter = __________

Perimeter = 12 in

Perimeter = __________

Perimeter = __________

Representation: Develop Language and Symbols. Invite students to represent each problem as an equation to help them identify strategies for solving and to give them practice interpreting mathematical language.
Supports accessibility for: Conceptual Processing, Language

Required Materials

Materials to Copy

• Missing Measurements - Large, Spanish
• Missing Measurements - Small, Spanish

Required Preparation

• If the activity is done as a gallery walk, print and cut 1–2 copies of the blackline master with the larger images and post them around the classroom. Otherwise, print and cut 1 copy of the blackline master with the smaller images for each group of 3–4 students.

Launch

• Groups of 3–4
• As a class, read the opening paragraph and the first two problems. Display the six diagrams. Give students a minute to ask clarifying questions.
• “Recuerden que los cuadriláteros son figuras de cuatro lados. Los cuadrados y los rectángulos son ejemplos de cuadriláteros, pero no son los únicos que hay” // “Recall that quadrilaterals are four-sided shapes. Squares and rectangles are examples of quadrilaterals, but they are not the only ones.”
• If done as a gallery walk, ask each student in a group to visit at least 2 posters, ensuring that, collectively, the members of each group visit all the posters.
• Alternatively, give each group one set of diagrams from the blackline master and ask each member to work on 2 of the diagrams.
• “Pónganse de acuerdo con su grupo para asegurarse de que contesten los seis problemas” // “Coordinate with your group to make sure that all six problems are answered.”

Activity

• 7–8 minutes: independent work time on the first problem
• 10 minutes: group work time on the remainder of the task
• Monitor for students who can distinguish customary units and standard units and consider them as categories of related units (for instance, yards, feet, and inches in one category, and kilometers, meters, and centimeters as another.)

Student Facing

Tu profesor colgó seis cuadriláteros alrededor del salón. Cada uno tiene una longitud de lado desconocida o un perímetro desconocido.

1. Escoge dos diagramas (uno que tenga una longitud desconocida y otro que tenga un perímetro desconocido). Asegúrate de que cada figura sea escogida por al menos una persona de tu grupo.

Encuentra los valores desconocidos. Muestra cómo razonaste y recuerda incluir las unidades.

2. Discute las respuestas con tu grupo hasta que todos estén de acuerdo en cuáles son las medidas desconocidas de las seis figuras.

3. Responde una de las siguientes preguntas. Explica o muestra cómo razonaste.

1. ¿El perímetro de la figura B es cuántas veces el perímetro de la figura D?
2. El perímetro de una figura es 1,000 veces el perímetro de otra figura. ¿Cuáles son las dos figuras?
3. ¿El perímetro de la figura F es cuántas veces el perímetro de la figura B?

Activity Synthesis

• See lesson synthesis.

Lesson Synthesis

Lesson Synthesis

Select students to briefly share the 6 missing measurements and their reasoning. Display their calculations or record them for all to see.

Then, discuss how students went about reasoning if the comparison statements in the last problem were true.

“¿Cómo supieron que el perímetro de la figura B es 2 veces el de la figura D?” // “How did you know that the perimeter of B is 2 times that of D?” (I know that feet and inches are related, and that 1 foot is 12 inches. So I recognized that 24 inches is twice 1 foot.)

“¿Cómo encontraron una figura que tuviera un perímetro que fuera 1,000 veces el perímetro de otra figura?” // “How did you find a figure with a perimeter 1,000 times that of another figure?” (I know that a kilometer is 1,000 meters, so I started with A and E, converted their perimeters, and then checked if one is indeed 1,000 times as long as the other.)

“¿Cómo supieron que el perímetro de la figura F era 9 veces el perímetro de la figura B?” // “How did you know that the perimeter of F is 9 times that of B?” (I converted the yards into feet, and then I could see that 18 feet and 2 feet are related: $$18 = 9 \times 2$$.)

Cool-down: Un rectángulo y un trapecio (5 minutes)

Cool-Down

• Una forma de resolver este problema es encontrar $$16 \div 4$$, que es la distancia en yardas del lanzamiento de Jada ($$16 \div 4 = 4$$), y luego multiplicar el resultado para convertir las yardas a pies ($$4 \times 3 = 12$$, así que 4 yardas son 12 pies).
• Otra forma de resolverlo es primero convertir las 16 yardas a pies ($$16 \times 3 = 48$$, así que 16 yardas son 48 pies) y luego dividir el resultado entre 4 para encontrar la distancia del lanzamiento de Jada ($$48 \div 4 = 12$$).