Lesson 7

Students commonly think that angles formed by longer segments are greater in size than those formed by shorter segments. The purpose of this warm-up is to bring up and address this likely misconception. The diagrams prompt students to observe the lengths of segments forming the angles and consider how they affect our perception of the size of the angles.

While students may notice and wonder many things about these sets of angles, it is important to discuss the relative sizes of the angles in the two sets. Make sure students see that the two sets of angles are identical in size even though the segments that form them seem to suggest otherwise.

Consider using patty paper to demonstrate equal-size angles during the synthesis.

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

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

• “¿En el grupo de arriba los ángulos se van haciendo más grandes o más pequeños? ¿Y qué pasa en el grupo de abajo?” // “Are the angles getting larger or getting smaller in the top set? What about in the bottom set?” (They are getting larger as you move from left to right.)
• “Comparen el primer ángulo del grupo 1 con el primero del grupo 2. ¿Qué pueden decir?” // “How does the first angle in set 1 compare to that in set 2?” (They look very similar and almost the same size.)
• “Comparen el segundo ángulo del grupo 1 con el segundo del grupo 2. ¿Qué pueden decir?” // “How does the second angle in set 1 compare to that in set 2?” (They look the same size but the rays that create the angles are longer in set 1 than in set 2.)
• “¿Qué pueden decir al comparar el tercer ángulo de un grupo con el tercer ángulo del otro?” // “What about the third angle in each set?” (Set 1 has shorter lines than set 2 but the angle is the same size.)
• “¿Cómo podemos decidir si uno es más grande, más pequeño o del mismo tamaño que el otro?” // “How can we find out if one is larger, smaller, or the same size as the other?” (Measure them by laying them on top of each other.)
• Consider using patty paper to trace corresponding angles in the two sets and show that they are the same size even if the segments in the second set are shorter.
• Highlight that the size of an angle is not determined by the length of the segments that frame it.

In a previous lesson, students described an angle to a peer so that they might draw the angle accurately without seeing it. Students learned what an angle is and have reasoned about how to describe the size of an angle.

In this activity, students use the features of an analog clock (minute hand, hour hand, and position of numbers) to explain how to draw a given angle. In doing so, they use language related to turning one or both hands on the clock. In future activities, students will relate this idea to turning a ray around a fixed point. 

The hands on an actual clock are designed so that moving the minute moves the hour hand. The examples here disregard this constraint. For the purposes of this activity, it is assumed that both hands can be moved freely, or that one hand can be held in place while the other moved freely.

• Groups of 2
• Give students access to rulers or straightedges.

• “En una lección anterior, jugamos un juego en el que le describían un ángulo a su compañero para ayudarlo a que lo dibujara sin verlo” // “In an earlier lesson, we played a game where you described an angle to your partner to help them draw it without seeing it.”
• “Lean la explicación de Andre sobre cómo dibujar un ángulo. Intenten dibujar su ángulo y luego piensen en otra manera de describir cómo dibujar el mismo ángulo” // “Read Andre’s explanation for how to draw an angle. Try to draw his angle, and then think of another way to describe how to draw the same angle.”
• 3 minutes: independent work time on the first two problems
• “Compartan su dibujo y su descripción con su compañero” // “Share your drawing and description with your partner.”
• “Discutan de qué manera explicarían cómo dibujar los otros ángulos” // “Discuss how you would explain how to draw the other angles.”
• 3–5 minutes: partner discussion and work time
• Monitor for students who clearly describe turning a hand around the clock and who connect the hands to the rays of the angles.

• “Escuchemos algunas instrucciones diferentes sobre cómo dibujar estos ángulos. Mientras escuchan, intenten dibujar el ángulo y adivinen cuál se está describiendo” //  “Let’s listen to some different directions for how to draw these angles. As you listen, try to sketch the angle and guess which one they are describing how to draw.”
• Invite 2–3 students to share the directions they wrote for drawing one of the angles. Choose students to share in a different order than the angles are presented in the task.
• For each, consider asking:
  • “¿Cuál ángulo describieron ellos (a, b o c) y cómo lo saben?” // “Which angle did they describe (a, b, or c) and how do you know?”
  • “¿Qué palabras les ayudaron a imaginarse cómo dibujar el ángulo?” // “What words help you picture how to draw the angle?” (turn right, turn left, point at 4)
• “Podemos describir el tamaño de un ángulo explicando cuánto giró un rayo desde el otro rayo” // “We can describe the size of an angle by explaining how much one ray has turned from the other.” 

In this activity, students compare the size of angles by thinking in terms of a turn from one ray from the other ray. They continue to use the clock as a tool for reasoning and for talking about “how much” of a turn. This work helps to elicit a need for more formal units and tools for measuring degrees while building the foundation for understanding angle measurement as additive. Students may describe the unit of measurement as “minutes,” “hours,” or other informal names (turn-units). They will learn about degrees as a unit of measurement in the upcoming lessons. 

The final question gives students an opportunity to describe the size of an angle in as many ways as they can, using the clockface, benchmark angles, or other informal descriptions (MP6) preparing them for the introduction of a numerical way to measure angles.

• Groups of 2
• “Antes usamos las manecillas de un reloj para describir cómo dibujar un ángulo” // “Earlier, we used the hands of a clock to describe how to draw an angle.”
• “Ahora pensemos en cómo un reloj nos podría ayudar a hablar sobre el tamaño de un ángulo” // “Now, let’s think about how a clock might help us talk about the size of an angle.”

• “En cada caso, usen los relojes para comparar los ángulos formados por las manecillas del reloj” // “Use the clocks to compare the angles formed by each clock’s hands.”
• 3–4 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for different ways students compare the same pairs of angles.

• Invite 2–3 previously identified students to share their responses. 
• “¿Cómo pudo cada estudiante ver el ángulo formado por las manecillas de una manera diferente?” // “How did each student see the angle formed by the hands in different ways?” (Some thought about moving the _____ hand toward the _____ hand, like the direction it really moves on the clock. Others thought about the minute hand starting at the same spot as the hour hand and moved it in either direction.)
• “¿Esto cómo cambió cuál era el ángulo que ellos pensaban que era el más grande?” // “How did this change which angle they thought was larger?” (It changed how far they thought one ray turned.)
• “Cuando le describimos un ángulo a otras personas, con frecuencia dibujamos un arco, o una parte de un círculo, entre los rayos para mostrar a cuál giro nos referimos” // “When we describe an angle to others, we often draw an arc, or part of a circle, between the rays to show which turn we are talking about.”