Lesson 9
Use a Protractor to Measure Angles
Warmup: True or False: There's Something about 45 (10 minutes)
Narrative
Launch
 Display one equation.
 “Give me a signal when you know whether the equation is true and can explain how you know.”
 1 minute: quiet think time
Activity
 Share and record answers and strategy.
 Repeat with each statement.
Student Facing
Decide if each statement is true or false. Be prepared to explain your reasoning.
 \(2 \times 45 = 6 \times 15\)
 \(4 \times 45 = 2 \times 90\)
 \(3 \times 45 = 180  90\)
 \(6 \times 45 = 45 + 90 + 135\)
Student Response
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Activity Synthesis
 Some students may notice that it is handy to think in terms of \(2 \times 45\) because it would mean dealing with multiples of 90 rather than multiples of 45. Highlight their explanations.
 If no students decomposed expressions such as \(3 \times 45\), \(4 \times 45\), and \(6 \times 45\) into sums of \(1 \times 45\) and \(2 \times 45\), discuss how this could be done. (See sample responses.)
Activity 1: How Large is a $1^\circ$ Angle? (15 minutes)
Narrative
This activity develops an understanding of the degree as a unit used to measure angles and introduces students to the protractor.
By now students have encountered many angle measurements and have some intuitive awareness of the sizes relative to a full turn (\(360^\circ\)), half of a full turn (\(180^\circ\)), and a quarter of a full turn (\(90^\circ\)). In this activity, students learn that a \(1^\circ\) angle is \(\frac{1}{360}\) of a full turn and that an angle that is composed of \(n\) \(1^\circ\) angles has a measurement of \(n^ \circ\). For example, a \(7^\circ \) angle is made up of seven \(1^\circ\)angles.
Advances: Conversing, Reading
Required Materials
Materials to Gather
Launch
 Groups of 2
 2 minutes: independent work time to find the fractions of a full turn
 Record students’ responses. Solicit some ideas on how large a \(1^\circ\) angle is. Emphasize that a \(1^\circ\) angle is \(\frac{1}{360}\)of a turn and that it’s the size of the angle formed by the sides of the pieces of the circle created if we cut a full circle into 360 equal parts.
 “To measure angles in degrees, we can use a protractor.” (Consider displaying different types of protractors.)
 Give each student a protractor.
 “Compare this tool to the one you used in the previous lesson. How are they the same? How are they different?” (They are both semicircles. They both show angles like \(0^\circ\), \(30^\circ\), \(45^\circ\), and \(90^\circ\). The protractor is transparent or has a hole, while the paper version is solid. The protractor shows many more lines or tick marks and more numbers around the curve.)
Activity
 2–3 minutes: group work time on the second set of questions.
 Monitor for students who look for structure to find the number of \(1^\circ\) increments on the protractor (for example, noticing that every group of ten \(1^\circ\) increments are marked and counting those instead of counting individual tick marks).
 Pause for a discussion. Make sure students see that a \(1^\circ\) angle on the protractor results when we draw rays through a pair of neighboring tick marks.
 2–3 minutes: individual or group work time on the remaining questions
Student Facing
 A ray that turns all the way around its endpoint and back to its starting place has made a full turn or has turned \(360^\circ\).
What fraction of a full turn is each of the following angle measurements?

\(120^\circ\)

\(60^\circ\)

\(45^\circ\)

\(30^\circ\)

\(10^\circ\)

\(1^\circ\)


Your teacher will give you a protractor, a tool for measuring the number of degrees in an angle.
 How is \(1^\circ\) shown on the protractor?
 How many \(1^\circ\) measurements do you see?

A protractor with no numbers has been placed over an angle.

The center of the protractor is lined up with the vertex of the angle.

The straight edge of the protractor is lined up with a ray of the angle.
How many degrees is this angle? Explain how you know.


An angle contains thirty \(1^\circ\) angles, as shown. How many degrees is this angle?
Student Response
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Activity Synthesis
 “How did you find out the size of the two angles when the protractors show no numbers or scales?” (We know that the turn from one tick mark to the next is \(1^\circ\). We can use the tick marks to count the number of \(1^\circ\) turns. We can imagine the tick marks split the angle into a number of \(1^\circ\) angles. We can count each \(1^\circ\)angle to find the measurement.)
Activity 2: Use a Protractor (20 minutes)
Narrative
In this activity, students learn how to use a protractor. They align a protractor to the vertex and a ray of an angle so that its measurement can be read. The given angles are oriented in different ways, drawing students’ attention to the two sets of scales on a protractor. Students need to consider which set of numbers to pay attention to and think about a possible explanation for when or how to use each scale. Moreover, one of the scales is only marked in increments of 10 degrees so if students use this scale they need to reason carefully about the precise angle measure (MP6).
This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.
Supports accessibility for: Conceptual Processing, VisualSpatial Processing, Attention
Launch
 Groups of 2–4
Activity
 5 minutes: independent work time
 2–3 minutes: group discussion
 Monitor for students who find each measurement by:
 looking for the scale with a ray on \(0^\circ\) and counting up on that scale
 subtracting the two numbers (on the same scale) that the rays pass through
MLR1 Stronger and Clearer Each Time
 “Share your response to the last problem with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
 3–5 minutes: partner discussion
 Repeat with 2–3 different partners.
 “Revise your initial draft based on the feedback you got from your partners.”
 2–3 minutes: independent work time
Student Facing

Here are four angles whose sizes you may have estimated earlier. A protractor has been placed over each angle. Measure the size of each angle in degrees.


Elena and Kiran are measuring an angle with a protractor. Elena says the angle is \(80^\circ\). Kiran says it shows \(100^\circ\). Why might they end up with different measurements? Which one is correct? Explain your reasoning.
Student Response
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Activity Synthesis
 Select students who used different strategies to share their responses and reasoning.
 Display the image of Elena and Kiran’s angle. If no students mentioned that \(80^\circ\) is not possible because the angle clearly appears greater than a right angle (\(90^\circ\)), consider asking how Elena can use what she knows about right angles to think about her measurement.
Lesson Synthesis
Lesson Synthesis
“Today, we used \(1^\circ\) angles and a protractor to measure the size of angles.”
“What do you know about a \(1^\circ\) angle?” (It’s \(\frac{1}{360}\) of a full turn of a ray through a circle. It's a small angle compared to the other angles we have seen. It’s \(\frac{1}{180}\) of a half turn of a ray through a circle, like on a protractor.)
“How can \(1^\circ\) angles tell us the size of other angles?” (It’s a smaller angle, so we can use it to be more precise when we measure or compare angles. We can count or find the number of \(1^\circ\) angles in an angle to find its measurement.)
“How would we know how many \(1^\circ\) angles are in another angle?” (We can use a protractor, which is marked with 180 or 360 onedegree angles.)
Display:
“We saw that a protractor has two sets of numbers. How do you know which set of numbers to use when measuring an angle?” (Either set could be used, but it is easier to use the set that counts up from 0 rather than count down from 180.)“Take 1–2 minutes to add any new words from today’s lesson to your word wall. Share your new entries with a neighbor and add any new ideas you learn from your conversation.”
Cooldown: Measure the Angles (5 minutes)
CoolDown
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