Lesson 6

This warm-up prompts students to compare four angles. It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the angles in comparison to one another. This language will be important as students sort and order angles in the lesson activities. During the synthesis, ask students to explain the meaning of any terminology they use, such as angle, ray, point, smaller, larger, flat, upside-down, and so on.

• Groups of 2
• Display the image.
• “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

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “¿Cada una de las imágenes muestra ángulos? ¿Cómo lo saben?” // “Do the images each show angles? How do you know?” (They each have two rays that share an endpoint. The angle in A doesn’t show arrows, but the line segments are on rays that start at the vertex.)
• “Encontremos al menos una razón por la que cada una es diferente” // “Let’s find at least one reason why each one doesn’t belong.”

Previously, students learned that angles are geometric figures made up of two rays that share a common endpoint. The purpose of this activity is for students to compare angles using the language that makes sense to them. As students look for ways to sort the cards into different categories, they have a reason to look for and describe the parts of the angles that make them different (MP7). Students may describe:

• the orientation of the rays
• the length of the segments used to represent the rays
• the point that the rays share
• the distance or “space” between the rays or other invented descriptions of the size of the angles

MLR2 Collect and Display is used during the synthesis of the lesson to record and organize the language students use to describe parts of angles and their size. Encourage students to notice differences in the ways they attempt to compare the angles. This work helps elicit the need for more precise vocabulary to describe angle size and ways to measure angles.

Here are images of the cards for reference:

This activity uses MLR2 Collect and Display. Advances: conversing, reading, writing.

• Create one set of cards from the blackline master for each group of 2 students.

• Groups of 2
• Give each group a set of cards from the blackline master.

MLR2 Collect and Display

• “Con su compañero, clasifiquen estos ángulos en distintas categorías. Expliquen las categorías que crearon” // “With your partner, sort these angles into different categories. Explain the categories you made.”
• Display the chart of words used to describe angles from the previous lesson.
• “Mientras piensan en cómo clasificar los ángulos, puede que les ayude usar el póster que creamos” // “As you think about how to sort the angles, it may be helpful to use the chart we created.”
• 3–5 minutes; group work time
• As students work, listen for and collect the language they use to add to the chart.
• Record students’ words and phrases on a visual display and update it throughout the lesson.
• “Ahora, van a comparar la forma como clasificaron los ángulos con la forma como lo hizo otro grupo” // “Now, you’ll compare how you sorted the angles and how another group did it.”
• “Cuando estén con el otro grupo, tómense un minuto para mirar las tarjetas del otro grupo sin hablar. Después, háganles las preguntas que tengan sobre sus categorías” // “When you have partnered up with another group, take a minute to look at their cards without talking. Then ask them questions you have about their categories.”
• Consider posting and encouraging students to use sentence starters such as:
  • “Observé que ustedes . . .” // “I noticed you . . . .”
  • “¿Ustedes cómo . . . ?” // “How did you . . . ?”
  • “¿Por qué ustedes . . . ?” // “Why did you . . . ?”
• Arrange for each group of 2 to work with another group.
• 1 minute: quiet think time
• 2–3 minutes: group discussion
• Monitor for a variety of ways students sort and explain their sorting decisions.

• “¿En qué fue parecida la manera de clasificar del otro grupo a la de ustedes?” // “How was others’ way of sorting like your group way of sorting?” (We all looked at the length of the drawn rays.) 
• “¿En qué fue diferente la manera de clasificar del otro grupo a la de ustedes?” // “How was their way of sorting different from yours?” (They sorted based on if a ray was pointing horizontally or not. We sorted based on if the rays were open wide or whether they looked like they were closing.)
• “¿Qué palabras usaron para ponerles un nombre a las categorías que crearon? ¿Qué palabras escucharon que otros usaron?” // “What words did you use to name the categories you created? What words did you hear others use?”
• As students share, update the display of words students use to describe the angles.
• Remind students to borrow language from the display as needed.

Previously, students sorted angles in ways that made sense to them. In this activity, students reason about how to compare angles based on a measurable attribute. They are asked to sort the angles from smallest to largest. Students may interpret this prompt in many ways, but all students must begin to reason about how to describe the size of an angle. As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).

Accept any way that students compare the angles so long as they can explain how they determine whether an angle is smaller or larger than another. Students may use informal language to describe the size of the angles. For example, they may use “less wide”, “more pointy”, or “less space” to describe smaller angles. They may also reason that an angle that “can fit inside” another angle is smaller. Continue to collect student language during the activity and to offer formal language when appropriate.  

• Groups of 2
• Make sure each group has the angle cards from the previous activity.
• Make patty paper available, if requested.

• “Ahora comparemos los ángulos que usamos en la actividad anterior de otra manera” // “Now let’s compare the angles we used in the last activity in a different way.”
• “Ordenen los ángulos del más pequeño al más grande. Discutan con su compañero qué piensan que significa eso. Después, ordenen las tarjetas” // “Order the angles from smallest to largest. Discuss with your partner what you think that might mean. Then, order the cards.”
• 5–7 minutes: partner work time
• As students work, monitor for the language they use and the features of the angles they focus on when ordering their size.

Some students may focus on the length of the line segments used to represent the rays. This is acceptable for the purposes of this activity, but encourage students to explain what they mean. Check that they understand that rays extend forever in one direction.

• Invite 2–3 groups who ordered the angles in different ways to share.
• For each, consider asking:
  • “¿En qué características del ángulo se enfocaron?” // “What features of the angle did you focus on?”
  • “¿Cómo decidieron si un ángulo era más pequeño o más grande?” // “How did you determine whether an angle was smaller or larger?”
• “Hay una manera estándar de medir el tamaño de los ángulos” // “There is a standard way of measuring the size of angles.”
• Display angles organized from smallest to largest: L, B, G, H, C, D, A (and F, K, P), O, J, E, M, N, I.
• “Estos ángulos están ordenados de la medida de ángulo más pequeña a la medida de ángulo más grande” // “These angles are ordered from the smallest angle measure to the largest angle measure.”
• “¿En qué se parece esta forma de ordenar los ángulos a como ustedes lo hicieron? ¿En qué se diferencia?” // “How is this the same as how you ordered the angles? How is it different?” (The smallest is the same in both. The largest are different, and the ones in middle are also different.)