Lesson 13

This warm-up prompts students to visualize the idea of arranging angles around a point and adding their measurements as more pieces are added. The angles are familiar angles: \(90^\circ\), \(180^\circ\), and \(270^\circ\). Students previously arrived at these benchmarks by decomposing a full turn or \(360^\circ\). Here, they compose a full turn from \(90^\circ\) angles.

The work here familiarizes students with the context and mathematics that might be involved later in the lesson. In the subsequent activities, students will compose and decompose paper cutouts of angles to determine angle measurements.

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

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

• “¿Cuál creen que es la medida de cada uno de los ángulos?” // “What angles do you think are the measurement of each angle?” (They look like \(90^\circ\), \(180^\circ\), and \(270^\circ\) angles.)
• “¿Cómo lo saben?” // “How do you know?” (If the paper is a rectangle, then the corner pieces are right angles or \(90^\circ\) each. Two of the corner pieces would be \(90+90\). Three pieces would be \(90+ 90+90\).)
• “¿Qué ángulo obtenemos si agregamos la última pieza de la esquina?” // “What angle would we get if we add the last corner piece?” (\(360^\circ\))

In this activity, students use their knowledge of \(90^\circ\), \(180^\circ\), and \(360^\circ\) and paper cutouts of some acute angles to determine the measurements of those angles. They then use those measurements to compose and find the measurements of larger angles.

No explicit directions for finding the angles are given, so students have an opportunity make sense of the problem and use what they know about the additivity of angles to find the angle measures (MP7). If requested, give students access to rectangular sheets of paper whose square corners could be torn off.

This activity uses MLR5 Co-Craft Questions. Advances: Reading, Writing.

• Create 4 copies of each angle (\(p\), \(q\), \(r\), and \(s\)) from the blackline master for each group of 2–4 students.
• Cut out the angles in advance, or prepare scissors and extra time for students to cut out the angles.
• If using patty paper instead of cutouts of the angles, each student needs 1–2 sheets of patty paper.

• Groups of 2–4
• Give each group the cutouts of the four angles, 4 copies of each angle per group, or if using patty paper, give 1–2 sheets to each student.
• If using patty paper, demonstrate that it can be used for tracing the angles.

MLR5 Co-Craft Questions

• Display only the image to the first problem without revealing the question.
• “Escriban una lista de preguntas matemáticas que se podrían hacer acerca de esta imagen” // “Write a list of mathematical questions that could be asked about this image.” (What is the size of each of the angles? Could we put angles together to make right angles? Would all of the angles make a straight line? Which angle is closest to 90 degrees and how far away from 90 degrees is it?)
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “¿En qué se parecen estas preguntas? ¿En qué son diferentes?” // “How are these questions alike? How are they different?” (Most questions are related to the size of the angles.)
• Reveal the task (students open books), and invite additional connections.
• Monitor for groups who:
  • compose 2 copies of angle \(p\) or 3 copies of angle \(q\) into a right angle
  • compose 2 copies of angle \(r\) into \(q\)
  • compose angles \(p\) and \(q\) into angle \(s\)
• Pause for a whole-class discussion. Select students or groups who reasoned as previously outlined to share their reasoning.
• 3–4 minutes: independent work time on the second question
• 2 minutes: group discussion
• Monitor for the different ways that smaller angles are used to compose the angles in parts a–d.

Offer patty paper to students who may need support using smaller angles to find the size of larger angles. Consider asking: “¿Cómo podría ayudarte trazar los ángulos a comparar los ángulos que conoces con los que no conoces?” // “How might tracing the angles help you compare the angles you know and those you don’t know?”

• Invite students to share their responses and reasoning for the second question. Display the angles.
• For each angle in parts a–d, record the different compositions of angles students use to find its measurement. For example:
  • To compose the angle in part a, students may use angles \(p\), \(r\), and \(s\), or they may use 3 copies of \(p\).
  • To find the angle in part c, students may use 3 copies of angle \(s\), or they may draw a line to separate \(180^\circ\) of the angle and fit angle \(p\) in the remaining space.

In this activity, students find the size of angles created by folding paper several times and reasoning about the resulting angles (MP7).

The first fold decomposes the paper into two congruent shapes whose edges line up exactly, and students could see how the folding split two of the angles into equal halves. The subsequent folds decompose an angle into two equal angles, but this may not be obvious to students because the shapes of the two resulting parts are different. (The edges or creases that form the angles are of different lengths.) Students need to remember that angles are not determined by the length of the segments that form it and reason accordingly.

Some students may need support in folding paper precisely. Consider providing a larger sheet of paper or a straightedge to facilitate the folding.

• Groups of 2–4
• Give each student a sheet of origami paper or square paper.
• Display the paper folding diagrams.

• 2–3 minutes: independent work time
• 2–3 minutes: group discussion
• Monitor for students who recognize that two angles are equal if the edges or creases that form them line up exactly (even if one crease or edge is longer than the other).

Students may benefit from reasoning about angle sizes concretely. Consider providing a piece of paper and asking after the first fold: “¿Cómo podemos saber cuál es el tamaño de los ángulos que hemos creado hasta este momento?” // “How might we tell the size of the angles we have created at this point?” and “¿Cómo podemos revisar lo que hemos pensado o mostrar que es correcto?” // “How can we check our thinking or show that it is correct?”

• Invite students to share their responses and reasoning.
• “¿Cómo podemos saber si se formaron parejas de ángulos del mismo tamaño con el primer doblez?” // “How can we tell if the first fold resulted in pairs of equal-size angles?” (The two triangles are identical and match up exactly, which means the angles in the two halves are the same.)
• “Las figuras que se forman con el segundo y con el tercer doblez no son la misma. ¿Esos dobleces también producen ángulos del mismo tamaño?” // “The shapes that result from the second and third folds are not the same. Do those folds produce equal-size angles as well?” (Yes. The edges of the resulting angles match up exactly and meet at the same endpoint, so the angles are the same size.)
• “¿Cuál ángulo es más grande: \(b\) o \(e\)?” // “Which angle is larger, \(b\) or \(e\)?” (They are the same, both are \(45^\circ\). Angle \(b\) is half of 90. Angle \(e\) is twice \(22.5^\circ\).)