Lesson 11

In this warm-up, students practice estimating a reasonable angle measurement based on their knowledge of angles so far and their familiarity with clocks. Later in the unit, students will take a closer look at the angles in an analog clock and apply their understanding of angles to solve more sophisticated problems.

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time

• 1 minute: partner discussion
• Record responses.
• Draw an arc to show the clockwise turn of the minute hand from the hour hand (label the \(143^\circ\)measurement with an arc).
• “Sus estimaciones deben mostrar el tamaño de este ángulo en grados. Si lo necesitan, ajusten sus estimaciones” // “Your estimate should show the size of this angle in degrees. If you need to, revise your estimates.”
• As needed, record any revisions.

• Consider asking:
  • “¿Alguien hizo una estimación menor que _____? ¿Alguien hizo una estimación mayor que _____?” // “Is anyone’s estimate less than _____? Is anyone’s estimate greater than _____?”
  • “¿Cómo lograron hacer una estimación? ¿Cómo supieron que _____ sería muy baja y _____ sería muy alta?” // “How did you go about making an estimate? How did you know that _____ must be too low and _____ must be too high?”
  • “Teniendo en cuenta esta discusión, ¿alguien quiere ajustar su estimación?” // “Based on this discussion does anyone want to revise their estimate?”
• Consider revealing the actual measurement: \(143^\circ\).

In this activity, students follow directions for drawing lines, rays, and angles. To create angles precisely and as specified, students need to use a protractor and a ruler or straightedge (MP6). 

Each step in the drawing process involves one or more decisions for students to make. In some cases, the resulting drawing will be the same.

For example, in the first question, students could use the protractors in different ways to create perpendicular lines.

Going from \(0^\circ\) to \(90^\circ\)

(outer set of numbers):

Going from \(20^\circ\) to \(110^\circ\)

(outer set of numbers):

Going from \(40^\circ\) to \(130^\circ\)

(inner set of numbers):

In other cases, the resulting drawings will vary depending on the decisions made. For example, in the second question, students could choose to draw the first angle (\(40^\circ\)) above or below the given ray. When drawing the second angle (\(20^\circ\)), they could choose to draw it inside the \(40^\circ\) angle or adjacent to the \(40^\circ\) angle (and choosing one side or the other)—in both cases meeting the specifications. The answer to the last question will thus also vary.

• Groups of 2
• Give each student a protractor and access to rulers or straightedges.
• 2 minutes: independent work time on the first question
• Pause for class discussion. Ask 1–2 students to share how they drew their perpendicular lines. 

• 5–6 minutes: independent work time on the remaining questions
• 2 minutes: partner discussion
• Identify students with different-looking drawings to share later.

• Select students to share their drawings and their reasoning for the last question.
• “¿Qué decisiones tuvieron que tomar al crear el dibujo?” // “What decisions did you have to make when creating the drawing?”
• “Muchos de ustedes ubicaron el ángulo de \(20^\circ\) al lado del ángulo de \(40^\circ\). Algunos de ustedes lo ubicaron dentro del ángulo de \(40^\circ\). ¿Cómo influyeron las distintas elecciones en el tamaño del ángulo de la última pregunta?” // “Many of you placed the \(20^\circ\) angle next to the \(40^\circ\) angle. Some of you placed them inside the \(40^\circ\) angle. How did the different choices affect the size of the angle in the last question?” (Putting the \(20^\circ\) inside the \(40^\circ\) angle made the last angle \(20^\circ\) larger.)

In the first activity, students drew angles with some scaffolding in place: a line and a point were given, each step was described, and the vertex and measurements of each angle were specified.

In this activity, students continue to draw angles but with less guidance. For each drawing, students are given only a range of angle measurements and no other criteria, prompting them to make additional decisions about how to draw the angles (for instance, where the vertex of an angle should be, how the first ray or line should be oriented, and so on). After drawing, students trade their cards and use a protractor to measure and check one another’s angles.

The drawings created here will be used in the next lesson. Consider collecting the cards from each group or otherwise supporting students in keeping the cards until then.

• Groups of 2
• Give each student one protractor and 4 blank (unlined) index cards.
• Give students access to rulers or straightedges.

• 7–8 minutes: independent work time on the first question, then switch cards and complete the second set of questions 

• Invite students to share 1–2 examples of an angle that meets each requirement.
• Consider asking:
  • “¿Pueden saber con solo mirar que este ángulo mide _____?” // “Can you tell just by looking that this angle is _____?”
  • “Si dijeron que sí, expliquen” // “If you say yes, explain.”
  • “Si dijeron que no, ¿qué necesitarían para asegurarse de que mide _____?” // “If you say no, what would you need to make sure it is _____?”