Lesson 14

This warm-up prompts students to carefully analyze and compare geometric features of four clock faces. Students may compare the times being represented, but because no numbers are shown, they are likely to compare the hands of the clocks and the angles they form.

In making comparisons, students have a reason to use language precisely (MP6). Teachers have a chance to hear the terminologies students use and how they talk about characteristics of angles.

• Groups of 2
• Display the image.
• “Escojan uno 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
• 2–3 minutes: partner discussion
• Record responses.
• As students explain, find opportunities to reinforce the terms “agudo” // “acute” and “obtuso” // “obtuse.”

• Consider stating: “Encontremos al menos una razón por la que cada uno es diferente” // “Let’s find at least one reason why each one doesn’t belong.”
• Display the following sentences and ask students to complete each sentence with “siempre” // “always,” “a veces” // “sometimes,” or “nunca” // “never:”
  • “La manecilla de las horas y la de los minutos de un reloj _____ forman un ángulo” // “The hour and minute hands of a clock _____ form one angle.”
  • “La manecilla de las horas y la de los minutos de un reloj _____ forman dos ángulos” // “The hour and minute hands of a clock _____ form two angles.”
• Ask students to explain their choice or use counterexamples to disagree with a classmate’s choice.

In an earlier lesson, students had folded paper and used supplemental tools to form and draw some benchmark angles (\(30^\circ\), \(45^\circ\), \(60^\circ\), and so on). In this activity, they apply their ability to measure and draw angles with a protractor to create a reasonably accurate clock face. The measuring and drawing here prepare students to reason about the angles formed by the hands of a clock in the next activity.

Students may notice that lines that give the positions of 1 and 2 on the clock can be extended through the center of the clock to give the positions of 7 and 8. Students who use these observations to create the drawing practice making use of structure (MP7).

The clock that students draw in this activity can be a helpful reference in the next activity.

• Groups of 2
• Give one protractor and a straightedge or a ruler to each student.

• 5 minutes: independent work time
• 1–2 minutes: partner discussion
• Monitor for students who:
  • measure the angle needed to place each number (one at a time)
  • draw the lines to position the numbers 1, 2, 4, and 5 and then try to find a way to mirror them vertically to locate 11, 10, 8, and 7
  • extend the lines they drew to find the numbers 1 and 2 and use them to find 7 and 8, and do the same with 3 and 4

• Display the incomplete clock face. Select students to share their completed drawing and their drawing process. Sequence the presentation in the order shown in the monitoring notes.
• “¿Cómo encontraron el tamaño del ángulo formado entre el número 1 y el número 2?” // “How did you find the size of the angle formed between the number 1 and 2?” (Divide 90 by 3, or divide 180 by 6, or divide 360 by 12.)
• “¿El ángulo formado por dos rayos consecutivos siempre es \(30^\circ\)? ¿Cómo lo saben?” // “Is the angle formed by any two consecutive rays always \(30^\circ\)? How do you know?” (Yes, because the angle is always \(\frac{1}{12}\) of 360.)

In grade 3, students learned to tell and write time to the nearest minute and measure time intervals in minutes. They understand that moving from one number on the clock to the next means 5 minutes have elapsed. In this activity, students build on those understandings to solve problems about angles formed by the hands of a clock.

Many students would benefit from having a visual reference of a clock as they are solving these problems. Encourage them to use their clock drawing from the previous activity for support.

Some students may try to answer each question by drawing each indicated time and then measuring the angles formed by the hands. Ask them to consider finding the size of the angles by reasoning and without measuring. For example, ask: “¿Qué saben sobre el ángulo que se forma cuando una manecilla va del 12 al 3?, ¿del 12 al 1?” // “What do you know about the angle that is formed when a hand goes from 12 to 3? From 12 to 1?” This encourages students to use the structure of the clock and the equal parts the clock face is divided into by the numbers on the clock (MP7).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.

• Groups of 4
• Give students access to protractors.

• 5 minutes: independent work time on the first two sets of questions

MLR1 Stronger and Clearer Each Time

• “Compartan su respuesta a la segunda pregunta con un compañero. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta el momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your response to the second question with a partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 1–2 minutes: structured partner discussion
• Repeat with 2 different partners (other members of the group).
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
• 2 minutes: independent work time
• 3–4 minutes: quiet work time on the last two sets of questions

• Display a clock face. Invite students to share their responses to the first question and the last two.
• When discussing the first set of questions, highlight that—except at 12 o’clock— the positions of the hour and minute hands produce two angles—a larger angle and a smaller angle.
• Likewise, when discussing the third question, if no students mentioned that there are two possible times that meet the described constraint, bring it up.
• “¿Cómo encontraron el número de grados que gira la manecilla de los minutos en 10 minutos y en 1 minuto?” // “How did you find out the number of degrees the minute hand turns in 10 minutes and 1 minute?” (Ten minutes is twice 5 minutes, so it is twice \(30^\circ\), or \(60^\circ\). One minute is 10 minutes divided by 10, so it is \(60^\circ\) divided by 10, which is \(6^\circ\).)