Lesson 8

The purpose of this warm-up is to invite students to think about 360 in terms of related numbers—as a result of addition, subtraction, multiplication, or division. The reasoning done here will be helpful when students compose angles into a sum of \(360^\circ\) or decompose a \(360^\circ\) angle into smaller angles, particularly benchmark angles such as \(30^\circ\), \(60^\circ\), \(90^\circ\), and \(180^\circ\).

• Display the number.
• “¿Qué saben sobre el 360?” // “What do you know about 360?”

• 1 minute: quiet think time
• Record responses.
• If no students mentioned different ways to express 360, ask: “How could we express the number 360?” and “What do you know about the factors of 360?”

• Draw students’ attention to the factors of 360. “¿Cuáles son los factores de 360? ¿Cuántos hay?” // “What are the factors of 360? How many are there?”
• “El número 360 y sus factores son importantes cuando describimos ángulos. Descubramos por qué aparecen una y otra vez mientras estudiamos una nueva manera de describir y medir el tamaño de los ángulos” // “The number 360 and its factors are important when describing angles. Let’s find out why they show up again and again as we look at a new way to describe and measure the size of angles.”

In previous activities, students used the features of a clock to describe and compare angles. This activity introduces students to degree as a unit of measure.

Because one degree is much more abstract than one inch or one square inch tile, students are first introduced to 360 degrees as a full rotation of a ray around its endpoint. Students use this information to reason about other angle measurements. They may use their understanding of fractions of a circle to determine the sizes (MP7).

In the synthesis, students describe angle measurement as additive. Students will continue to build this understanding and reason about the size of 1 degree in subsequent lesson activities.

• Groups of 2
• “Así como la longitud y el área se pueden medir en unidades estándar, los ángulos también tienen unidades estándar” // “Just like length and area can be measured in standard units, angles have standard units.”
• “Una unidad estándar para medir el tamaño de los ángulos es el grado” // “One standard unit for measuring the size of angles is degree.”
• Display the image of the ray turning 360 degrees and read the opening task statement.

• “Usen esta información para encontrar cuántos grados giró cada rayo desde donde comenzó. Después, dibujen algunos ángulos que tengan aproximadamente el tamaño dado en grados” // “Use this information to determine how many degrees each ray has turned from where it started. Then sketch some angles that are about the size given in degrees.”
• 3–4 minutes: independent work time
• “Comparen con su compañero cómo pensaron. Expliquen cómo hicieron sus estimaciones” // “Compare your thinking with your partner. Explain how you made your estimates.”
• 2–3 minutes: partner discussion
• Monitor for students who:
  • describe the 180 degree turn as half a full turn or turning half a circle
  • describe the 90 degree angle as half of half a turn or as a turning a fourth of the way around the circle
  • use their estimate for the 90 degree angle to estimate the 270 degree angle by adding (\(90 + 90 + 90\) or \(180 + 90\)) or by subtracting (\(360 - 90\))

• Invite previously identified students to share how they estimated the turn of each ray in degrees.
• “¿En qué se parecieron sus métodos? ¿En qué fueron diferentes?” // “How were your methods the same? How were they different?” (We all used the first diagram to figure out how much of a turn. Some of us used one measure to find the next measure. Some of us thought about addition and some thought about division).
• Consider displaying the equation \(90 + 90 + 90 = 270\) and discussing:
  • “¿Cómo corresponde esto a una manera de estimar el tercer ángulo?” // “How does this match a way to estimate the third angle?”
  • “¿Qué otra ecuación podemos usar para describir ese ángulo?” // “What’s another equation we could use to describe that angle?”
• “Un ángulo que mide 90 grados se llama un ángulo recto” // “An angle that measures 90 degrees is called a right angle.”
• “¿En dónde han visto ángulos rectos antes?” // “Where have you seen right angles before?” (Corners of squares or rectangles. Corners of paper. Angle made by the hands of a clock when it is 3 o'clock or 9 o'clock.)

In this activity, students construct a protractor-like tool that shows some benchmark angles. They do so by halving given angles—\(120^\circ\) and \(180^\circ\)—and then of subsequent angles identified along the way.

The activity serves several goals. The first is to familiarize students with the structure of a protractor using tactile processes (folding paper and aligning lines or edges). The second goal is to develop students’ intuition for thinking of a larger angle as composed of smaller angles, preparing them to see (in future lessons) that a \(1^\circ\) angle is \(\frac{1}{360}\) of a full turn. A final goal is to motivate the need for a tool that can measure angles more precisely.

Some students may use a square corner of a sheet of paper to find a \(90^\circ\) angle on their semi-circle and others may choose to fold their semi-circle in half. Expect most students to fold their paper to find all subsequent angles.

• Create a paper half-circle from the blackline master for each student.

• Groups of 2–4
• Give each student one paper half-circle and access to rulers or straightedges.
• “Su hoja de papel tiene la forma de medio círculo. La hoja muestra un rayo en la parte inferior derecha y dos ángulos (\(120^\circ\) y \(180^\circ\)) medidos desde el rayo” // “Your sheet of paper is in the shape of half a circle. It shows a ray on the bottom right and two angles (\(120^\circ\) and \(180^\circ\)) measured from the ray.“
• “Vemos la marca del \(120^\circ\). ¿Dónde está el ángulo de \(120^\circ\)? ¿Dónde están los dos rayos que forman este ángulo?” // “We see the \(120^\circ\) label. Where is the \(120^\circ\) angle? Where are the two rays that make this angle?”
• 1 minute: quiet think time for the first problem
• 1 minute: group discussion
• “¿Dónde piensan que estaría el segundo rayo de un ángulo de \(90^\circ\)?” // “Where do you think the second ray of a \(90^\circ\) angle would be?” (Between 0 and 120, but closer to 120.)

• 5–7 minutes: independent work time
• As students work on the last problem, monitor their ideas for using their tool to estimate angle measurements.

Students may create angles that are not precise when they estimate where to draw a new line segment. Ask the students to explain how they determine where to draw a line segment and suggest folding as a strategy. Consider asking:

• “¿Piensas que el ángulo que necesitas crear es más pequeño o más grande que los que ya dibujaste? ¿Cuánto más pequeño o más grande?” // “Do you think the angle you need to create is smaller or larger than those that you have already drawn? How much smaller or larger?”
• “Si hicieras dobleces, ¿cómo te ayudaría eso a crear ángulos más precisos en esta actividad?” // “How might folding help you create more precise angles during this task?”

• “¿Cómo encontraron un ángulo de \(90^\circ\)?” // “How did you find a \(90^\circ\) angle?” (I folded the semi-circle into halves.)
• “¿Cómo encontraron todos los demás ángulos?” // “How did you find all the other angles?” (For \(60^\circ\), fold to line up the thick ray with the \(120^\circ\) line, splitting 120 into two. For \(45^\circ\), fold to line up the ray with the \(90^\circ\) line, splitting 90 into two. Repeat in a similar fashion to find the others.)
• Invite students to share how they might find a \(1^\circ\) angle on their half circle. Highlight explanations that involve finding some fraction of increasingly smaller and smaller angles.
• Solicit some estimates of the angle measurements in the last problem. Record and display them for comparison later, when the same four angles are measured using a protractor.
• Students are likely to notice that their tool is imprecise and is not reliable or practical for measuring angles beyond estimations. Explain that we will look at another tool in the next activity.