Lesson 15

In this warm-up, students practice identifying obtuse angles in an image. They may, for instance, rely on the symmetry of the figure or on a grouping strategy, or otherwise scan the figure in a methodical way.

• Groups of 2
• “¿Cuántos ángulos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many angles do you see? How do you see them?”
• Display the image.
• 1 minute: quiet think time

• Display the image.
• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿Cómo se aseguraron de haber tenido en cuenta todos los ángulos?” // “How did you make sure all the angles are accounted for?”(I put a mark through them or numbered them.)
• “¿Cuántos ángulos obtusos hay en esta imagen?” // “How many obtuse angles are in this image?” (10)
• Label each obtuse angle with reasoning from students.
• Consider asking:
  • “¿Alguien puede expresar con otras palabras la manera en la que _____ vio los ángulos?” // “Who can restate in different words the way _____ saw the angles?”
  • “¿Alguien vio los ángulos de la misma manera, pero lo explicaría de otra forma?” // “Did anyone see the angles the same way but would explain it differently?”
  • “¿Alguien quiere compartir otra observación sobre la manera en la que _____ vio los ángulos?” // “Does anyone want to add an observation to the way _____ saw the angles?”

Previously, students found numerous angle sizes by reasoning and without using a protractor. They have done so with problems with and without context. In this activity, students consolidate various skills and understandings gained in the unit and apply them to solve problems that are more abstract and complex. They rely, in particular, on their knowledge of right angles and straight angles to reason about unknown measurements. (Students may need a reminder that an angle marked with a small square is a right angle.)

The angles with unknown measurements are shaded but not labeled, motivating students to consider representing them (or their values) with symbols or letters for easier reference. Students may also choose to write equations to show how they are thinking about the problems.

When students use the fact that angles making a line add up to \(180^\circ\) and that angles making a right angle add up to \(90^\circ\) they make use of structure to find the unknown angle measures (MP7).

• Groups of 2

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who:
  • use symbols or letters to represent unknown angles
  • write equations to help them reason about the angle measurements

To find the size of the shaded acute angle in the last diagram, students will need to reason indirectly by first finding the size of the shaded obtuse angle. (At this stage they are not expected to know that vertical angles are always equal.) Students who try to find the measurement of the acute angle first will likely get stuck. Consider asking students what they know about straight angles and how they can use what they know to pick which unknown angle to find first.

• Display the angles. Select students to share their responses. Record and display their reasoning.
• Highlight equations that illuminate the relationship between the known angle, the unknown angle, and the reference angle (\(90^\circ\), \(180^\circ\), or \(360^\circ\)). For instance: \(62 + p = 90\), \(71 + r + 90 = 180\), \(x + 154 = 180\), and so on.
• Label the diagrams with letters or symbols as needed to facilitate equation writing.
• When discussing the last question, highlight that finding unknown values sometimes involve multiple steps, and some steps may need to happen before others.

In this Info Gap activity, students solve abstract multi-step problems involving an arrangement of angles with several unknown measurements. By now students have the knowledge and skills to find each unknown value, but the complexity of the diagram and the Info Gap structure demand that students carefully make sense of the visual information and look for entry points for solving the problems. They need to determine what information is necessary, ask for it, and persevere if their initial requests do not yield the information they need (MP1). The process also prompts them to refine the language they use and ask increasingly more precise questions until they get useful input (MP6).

Here is an image of the cards for reference:

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

• Groups of 2

MLR4 Information Gap

• Display the task statement, which shows a diagram of the Info Gap structure.
• 1–2 minutes: quiet think time
• Read the steps of the routine aloud.
• “Yo les voy a dar una tarjeta de problema o una tarjeta de datos. Lean su tarjeta en silencio. No se la lean ni se la muestren a su compañero” // “I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.”
• Distribute the cards.
• “El diagrama no está dibujado con precisión, así que no les recomiendo usar un transportador para medir” // “The diagram is not drawn accurately, so using a protractor to measure is not recommended.”
• 1–2 minutes: quiet think time
• Remind students that after the person with the problem card asks for a piece of information, the person with the data card should respond with “¿Por qué necesitas saber ________?” // “Why do you need to know (restate the information requested)?”

• 5 minutes: partner work time
• After students solve the first problem, distribute the next set of cards. Students switch roles and repeat the process with Problem Card 2 and Data Card 2.

Some students may be overwhelmed by the visual information. Ask them to try isolating a part of the diagram at a time, covering other parts that are not immediately relevant.

• Select students to share how they found each angle measure. Record their reasoning and highlight equations that clearly show the relationships between angles.
• “¿Para cuáles ángulos fue fácil encontrar sus medidas? ¿Qué hizo que fuera fácil?” // “Which angle measurements were easy to find? What made them easy?” (Sample response: \(p\) and \(d\), because it was fairly easy to see that each of them and a neighboring angle make a straight angle.)
• “¿Cuáles fueron un poco más complicados? ¿Por qué?” // “Which ones were a bit more involved? Why?” (Sample response: \(e\), because there are 5 angles that meet at that point. We needed to find \(a\) or \(d\) before finding \(e\).)