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 likely will 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 terminology students use to talk about the characteristics of angles.
Launch
Groups of 2
Display the image.
“Escojan 3 relojes que vayan juntos. Prepárense para compartir por qué van juntos” // “Pick 3 clock faces that go together. Be ready to share why they go together.”
Activity
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.”
¿Cuáles 3 van juntos?
Student Response
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Advancing Student Thinking
Activity Synthesis
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 other students' choice.
Activity 1
Standards Alignment
Building On
Addressing
4.MD.C.6
Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
In an earlier lesson, students had folded paper and used supplemental tools to form and draw some benchmark angles (, , , 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, respectively. 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.
MLR8 Discussion Supports. Use multimodal examples to show the relationship between the angles and the numbers on a clock. Use verbal descriptions along with gestures, drawings, or concrete objects to show how to precisely complete a clock. Advances: Listening, Representing
Launch
Groups of 2
Give each student a protractor and a ruler or a straightedge.
Activity
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, respectively, and then do the same with 4 and 5 (and 10 and 11, respectively).
Kiran dibuja un reloj. Él dibuja un par de rectas perpendiculares para encontrar la ubicación de los números 3, 6, 9 y 12 alrededor del círculo.
¿Cuántos grados mide cada ángulo? Explica cómo lo sabes.
Ayuda a Kiran a encontrar la ubicación exacta de los números “1” y “2” en el reloj.
¿Cuántas nuevas rectas tiene que dibujar?
Describe los ángulos que se deberían formar entre las 2 rectas que él ya dibujó y las nuevas rectas.
Dibuja las rectas con precisión, y ubica los números “1” y “2” en el dibujo.
Mide y dibuja todas las rectas que sean necesarias para completar el dibujo del reloj. Asegúrate de que cada número esté ubicado en el lugar correcto en el reloj.
Student Response
Activity Synthesis
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 los números 1 y 2?” // “How did you find the size of the angle formed between the numbers 1 and 2?” (Divide 90 by 3, or divide 180 by 6, or divide 360 by 12.)
“¿El ángulo formado por dos rayos que señalan dos números consecutivos siempre es ? ¿Cómo lo saben?” // "Is the angle formed by rays that point at any two consecutive numbers always ? How do you know?” (Yes, because the angle is always of 360.)
Activity 2
Standards Alignment
Building On
Addressing
4.MD.C.7
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
In grade 3, students learned to tell and write time to the nearest minute and to 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 the questions 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, 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 into which the numbers divide the clock face (MP7).
This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.
Engagement: Develop Effort and Persistence. SStudents may benefit from feedback that emphasizes effort, time on task, and continuous learning. For example, invite students to choose which part of the first two questions to start, and let them know that they will have the opportunity to share and revise their thinking throughout the lesson. Share examples of students who revised their drafts after discussing with a partner. Supports accessibility for: Social Emotional Functioning
Launch
Groups of 4
Give students access to protractors.
Activity
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 hecho 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 work 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
La manecilla de las horas y la manecilla de los minutos forman un ángulo en cada una de estas horas. ¿Cuántos grados mide cada ángulo?
6 en punto
8 en punto
9 en punto
11 en punto
12 en punto
¿Cuántos grados gira la manecilla de los minutos cuando se mueve desde las 2:00 hasta las 2:05?
¿Y cuántos grados gira cuando se mueve desde las 2:05 hasta las 2:30? Explica cómo lo sabes.
La manecilla de los minutos del reloj está en posición vertical a las 7 p.m. Un poco más tarde, forma un ángulo de con la posición en la que estaba a las 7 p.m. ¿Qué hora es?
Descifra cuántos grados gira la manecilla de los minutos durante:
10 minutos
1 minuto
4 minutos
Activity Synthesis
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 the positions of the hour and minute hands produce two angles—a larger angle and a smaller angle. Once each hour, the hands form a straight angle, dividing the clock into two equal semicircles. Once each hour, the hands overlap each other, producing one 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 , or . One minute is 10 minutes divided by 10, so it is divided by 10, which is .)
Lesson Synthesis
“Hoy aprendimos acerca de las medidas de algunos ángulos que están en un reloj. Miramos ángulos que están formados por las dos manecillas y también pensamos en el número de grados que gira la manecilla de los minutos cuando pasa el tiempo” // “Today we learned about angle measurements on a clock. We looked at the angles formed by the two hands, and we also thought about the number of degrees that a minute hand turns over time.”
“Para encontrar el tamaño de un ángulo que está en un reloj ¿es más útil pensar en términos del número de minutos, del número de grupos de 5 minutos o de los números del 1 al 12?” // “Which is more useful for finding the size of angle on a clock: thinking in terms of the number of minutes, the number of 5 minutes, or the numbers 1–12?” (It depends on the situation.)
Display the following images of clocks:
“¿Cada minuto, la manecilla de los minutos de un reloj cuadrado o de un reloj ovalado gira el mismo número de grados que gira la manecilla de un reloj redondo? Expliquen o muestren cómo lo saben” // “Does the minute hand on a square clock or an oval clock turn the same number of degrees every minute as it does on a round clock? Explain or show how you know.” (Yes. The minute hand still travels a full turn or , in an hour, or 60 minutes, so each minute it still travels , regardless of the outer shape of the clock or how far away the numbers are from the center point.)
Consider displaying an image of the oval clock showing 12 equal angles. Reinforce the idea that the size of an angle is not determined by the lengths of the segments of the rays that form the angle.
“Tómense 1 o 2 minutos para agregar a su muro de palabras las palabras nuevas de las últimas dos lecciones. Compartan sus palabras nuevas con un compañero y agreguen las nuevas ideas que surjan de su conversación” // “Take 1–2 minutes to add the new words from the past two lessons to your word wall. Share your new entries with a neighbor, and add any new ideas you learn from your conversation.”
Standards Alignment
Building On
4.G.A.1
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
